| L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s − 3·5-s + 16·6-s − 20·8-s − 2·9-s + 12·10-s − 40·12-s + 13-s + 12·15-s + 35·16-s + 3·17-s + 8·18-s − 2·19-s − 30·20-s − 9·23-s + 80·24-s + 4·25-s − 4·26-s + 40·27-s − 18·29-s − 48·30-s + 19·31-s − 56·32-s − 12·34-s − 20·36-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s − 1.34·5-s + 6.53·6-s − 7.07·8-s − 2/3·9-s + 3.79·10-s − 11.5·12-s + 0.277·13-s + 3.09·15-s + 35/4·16-s + 0.727·17-s + 1.88·18-s − 0.458·19-s − 6.70·20-s − 1.87·23-s + 16.3·24-s + 4/5·25-s − 0.784·26-s + 7.69·27-s − 3.34·29-s − 8.76·30-s + 3.41·31-s − 9.89·32-s − 2.05·34-s − 3.33·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 3 T + p T^{2} - p T^{3} - 27 T^{4} - p^{2} T^{5} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + p T^{2} - 49 T^{3} + 15 T^{4} - 49 p T^{5} + p^{3} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - T + 25 T^{2} - 14 T^{3} + 434 T^{4} - 14 p T^{5} + 25 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 3 T + 26 T^{2} - 62 T^{3} + 354 T^{4} - 62 p T^{5} + 26 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 13 T^{2} + 63 T^{3} + 453 T^{4} + 63 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 9 T + 86 T^{2} + 440 T^{3} + 2670 T^{4} + 440 p T^{5} + 86 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 18 T + 215 T^{2} + 1689 T^{3} + 10572 T^{4} + 1689 p T^{5} + 215 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 19 T + 205 T^{2} - 1484 T^{3} + 8978 T^{4} - 1484 p T^{5} + 205 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 2 T + 55 T^{2} + 61 T^{3} + 2110 T^{4} + 61 p T^{5} + 55 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 5 T + 127 T^{2} - 696 T^{3} + 7236 T^{4} - 696 p T^{5} + 127 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 155 T^{2} + 37 T^{3} + 10176 T^{4} + 37 p T^{5} + 155 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 T + 215 T^{2} - 929 T^{3} + 17154 T^{4} - 929 p T^{5} + 215 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 3 T + 68 T^{2} - 322 T^{3} + 7560 T^{4} - 322 p T^{5} + 68 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - T + 73 T^{2} - 11 p T^{3} + 1181 T^{4} - 11 p^{2} T^{5} + 73 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 11 T + 4 p T^{2} - 2026 T^{3} + 26702 T^{4} - 2026 p T^{5} + 4 p^{3} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 9 T + 107 T^{2} + 199 T^{3} + 1653 T^{4} + 199 p T^{5} + 107 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 22 T + 409 T^{2} - 4787 T^{3} + 48782 T^{4} - 4787 p T^{5} + 409 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 28 T + 541 T^{2} + 7283 T^{3} + 73511 T^{4} + 7283 p T^{5} + 541 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 12 T + 125 T^{2} - 957 T^{3} + 3390 T^{4} - 957 p T^{5} + 125 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 6 T + 305 T^{2} + 1435 T^{3} + 39018 T^{4} + 1435 p T^{5} + 305 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 11 T + 298 T^{2} + 2922 T^{3} + 39354 T^{4} + 2922 p T^{5} + 298 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.46337231602413814271572380866, −6.21685303442716151984570413807, −5.97066168699358439221543830266, −5.95437681611862773551589732124, −5.80999341501500385649457455283, −5.32044000879375004178927068692, −5.28717319423912055591359939929, −5.24589097915252474853519487319, −5.16171693057054394901433430381, −4.64339742515691938973824524408, −4.30278543217610876008074764955, −4.06131339733101074637656689035, −3.99368335820389681544689855949, −3.58457596492053667160883960879, −3.57195435039982563561154046486, −3.21800418201100706804047573273, −2.78707552541691100782117226125, −2.64645190648153209248337556541, −2.48309003028046306092896901160, −2.34635153391877446392579763869, −2.19900369907783312195856341853, −1.37640815883171347068340344167, −1.21899276066915905405013564788, −1.05233127433709037980121236738, −0.842866331575908248295970157323, 0, 0, 0, 0,
0.842866331575908248295970157323, 1.05233127433709037980121236738, 1.21899276066915905405013564788, 1.37640815883171347068340344167, 2.19900369907783312195856341853, 2.34635153391877446392579763869, 2.48309003028046306092896901160, 2.64645190648153209248337556541, 2.78707552541691100782117226125, 3.21800418201100706804047573273, 3.57195435039982563561154046486, 3.58457596492053667160883960879, 3.99368335820389681544689855949, 4.06131339733101074637656689035, 4.30278543217610876008074764955, 4.64339742515691938973824524408, 5.16171693057054394901433430381, 5.24589097915252474853519487319, 5.28717319423912055591359939929, 5.32044000879375004178927068692, 5.80999341501500385649457455283, 5.95437681611862773551589732124, 5.97066168699358439221543830266, 6.21685303442716151984570413807, 6.46337231602413814271572380866
