L(s) = 1 | + 3·3-s − 7-s − 9-s + 4·11-s + 13-s − 5·17-s − 4·19-s − 3·21-s + 5·23-s − 13·27-s − 11·29-s − 4·31-s + 12·33-s − 6·37-s + 3·39-s − 8·41-s + 3·43-s + 2·47-s − 18·49-s − 15·51-s − 18·53-s − 12·57-s − 23·59-s − 26·61-s + 63-s + 3·67-s + 15·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.377·7-s − 1/3·9-s + 1.20·11-s + 0.277·13-s − 1.21·17-s − 0.917·19-s − 0.654·21-s + 1.04·23-s − 2.50·27-s − 2.04·29-s − 0.718·31-s + 2.08·33-s − 0.986·37-s + 0.480·39-s − 1.24·41-s + 0.457·43-s + 0.291·47-s − 2.57·49-s − 2.10·51-s − 2.47·53-s − 1.58·57-s − 2.99·59-s − 3.32·61-s + 0.125·63-s + 0.366·67-s + 1.80·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{5} \) |
good | 3 | $C_2 \wr S_5$ | \( 1 - p T + 10 T^{2} - 20 T^{3} + 44 T^{4} - 76 T^{5} + 44 p T^{6} - 20 p^{2} T^{7} + 10 p^{3} T^{8} - p^{5} T^{9} + p^{5} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + T + 19 T^{2} + 16 T^{3} + 30 p T^{4} + 158 T^{5} + 30 p^{2} T^{6} + 16 p^{2} T^{7} + 19 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 4 T + 38 T^{2} - 8 p T^{3} + 557 T^{4} - 960 T^{5} + 557 p T^{6} - 8 p^{3} T^{7} + 38 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - T + 28 T^{2} - 4 T^{3} + 568 T^{4} - 220 T^{5} + 568 p T^{6} - 4 p^{2} T^{7} + 28 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 + 5 T + 57 T^{2} + 224 T^{3} + 1390 T^{4} + 4758 T^{5} + 1390 p T^{6} + 224 p^{2} T^{7} + 57 p^{3} T^{8} + 5 p^{4} T^{9} + p^{5} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 4 T + 42 T^{2} + 40 T^{3} + 317 T^{4} - 1304 T^{5} + 317 p T^{6} + 40 p^{2} T^{7} + 42 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 11 T + 134 T^{2} + 970 T^{3} + 238 p T^{4} + 37502 T^{5} + 238 p^{2} T^{6} + 970 p^{2} T^{7} + 134 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 4 T + 80 T^{2} + 337 T^{3} + 3437 T^{4} + 12806 T^{5} + 3437 p T^{6} + 337 p^{2} T^{7} + 80 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 6 T + 45 T^{2} + 312 T^{3} + 2822 T^{4} + 12004 T^{5} + 2822 p T^{6} + 312 p^{2} T^{7} + 45 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 8 T + 153 T^{2} + 641 T^{3} + 8317 T^{4} + 23597 T^{5} + 8317 p T^{6} + 641 p^{2} T^{7} + 153 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 3 T + 71 T^{2} - 192 T^{3} + 4450 T^{4} - 13194 T^{5} + 4450 p T^{6} - 192 p^{2} T^{7} + 71 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 2 T + 68 T^{2} - 343 T^{3} + 3101 T^{4} - 29318 T^{5} + 3101 p T^{6} - 343 p^{2} T^{7} + 68 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 18 T + 213 T^{2} + 1184 T^{3} + 4462 T^{4} + 2236 T^{5} + 4462 p T^{6} + 1184 p^{2} T^{7} + 213 p^{3} T^{8} + 18 p^{4} T^{9} + p^{5} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 23 T + 348 T^{2} + 4341 T^{3} + 43633 T^{4} + 357240 T^{5} + 43633 p T^{6} + 4341 p^{2} T^{7} + 348 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 26 T + 541 T^{2} + 7192 T^{3} + 81278 T^{4} + 683676 T^{5} + 81278 p T^{6} + 7192 p^{2} T^{7} + 541 p^{3} T^{8} + 26 p^{4} T^{9} + p^{5} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 3 T + 275 T^{2} - 612 T^{3} + 33150 T^{4} - 55586 T^{5} + 33150 p T^{6} - 612 p^{2} T^{7} + 275 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 - 2 T + 288 T^{2} - 435 T^{3} + 36805 T^{4} - 41598 T^{5} + 36805 p T^{6} - 435 p^{2} T^{7} + 288 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 4 T - 3 T^{2} + 167 T^{3} + 6489 T^{4} + 34251 T^{5} + 6489 p T^{6} + 167 p^{2} T^{7} - 3 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 43 T + 1029 T^{2} + 16880 T^{3} + 210588 T^{4} + 2084442 T^{5} + 210588 p T^{6} + 16880 p^{2} T^{7} + 1029 p^{3} T^{8} + 43 p^{4} T^{9} + p^{5} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 30 T + 584 T^{2} - 8400 T^{3} + 102155 T^{4} - 1014020 T^{5} + 102155 p T^{6} - 8400 p^{2} T^{7} + 584 p^{3} T^{8} - 30 p^{4} T^{9} + p^{5} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 15 T + 291 T^{2} - 2152 T^{3} + 25484 T^{4} - 1554 p T^{5} + 25484 p T^{6} - 2152 p^{2} T^{7} + 291 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 8 T + 257 T^{2} + 1592 T^{3} + 32590 T^{4} + 159328 T^{5} + 32590 p T^{6} + 1592 p^{2} T^{7} + 257 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.96691131083385917689273542144, −4.72563150023134228655925247310, −4.62967575451057922550484188968, −4.40020763458026733438024368809, −4.27853678532119686972122076612, −3.97983289478178581665339290915, −3.78478213769613339121155031540, −3.74585084806336749990811393505, −3.74492555172337682154188545436, −3.58836814714899905798035659115, −3.33644928671371813432387614891, −3.04661028608949739293117156465, −3.02029655438905375156345443573, −2.99665871571898703249447848744, −2.83446121425663154290997399841, −2.50352912359213879421050080067, −2.46800617889514193498795896437, −2.28756547041995820724106127323, −2.08421668575476058887325700972, −1.75211439659088975883798626920, −1.60744166072743405195730721939, −1.57302079595254928923907505055, −1.29758010873196924005273091282, −1.25915992000865786896168219835, −1.05678379464766611789786529631, 0, 0, 0, 0, 0,
1.05678379464766611789786529631, 1.25915992000865786896168219835, 1.29758010873196924005273091282, 1.57302079595254928923907505055, 1.60744166072743405195730721939, 1.75211439659088975883798626920, 2.08421668575476058887325700972, 2.28756547041995820724106127323, 2.46800617889514193498795896437, 2.50352912359213879421050080067, 2.83446121425663154290997399841, 2.99665871571898703249447848744, 3.02029655438905375156345443573, 3.04661028608949739293117156465, 3.33644928671371813432387614891, 3.58836814714899905798035659115, 3.74492555172337682154188545436, 3.74585084806336749990811393505, 3.78478213769613339121155031540, 3.97983289478178581665339290915, 4.27853678532119686972122076612, 4.40020763458026733438024368809, 4.62967575451057922550484188968, 4.72563150023134228655925247310, 4.96691131083385917689273542144