| L(s) = 1 | − 2-s − 3-s + 6-s − 8·7-s − 2·11-s + 6·13-s + 8·14-s + 12·17-s + 8·21-s + 2·22-s + 6·23-s − 6·26-s − 12·31-s + 32-s + 2·33-s − 12·34-s − 13·37-s − 6·39-s − 12·41-s − 8·42-s − 4·43-s − 6·46-s + 2·47-s + 12·49-s − 12·51-s + 21·53-s + 20·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 3.02·7-s − 0.603·11-s + 1.66·13-s + 2.13·14-s + 2.91·17-s + 1.74·21-s + 0.426·22-s + 1.25·23-s − 1.17·26-s − 2.15·31-s + 0.176·32-s + 0.348·33-s − 2.05·34-s − 2.13·37-s − 0.960·39-s − 1.87·41-s − 1.23·42-s − 0.609·43-s − 0.884·46-s + 0.291·47-s + 12/7·49-s − 1.68·51-s + 2.88·53-s + 2.60·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7015409457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7015409457\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 11 | $C_4\times C_2$ | \( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 34 p T^{5} + 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 12 T + 37 T^{2} + 180 T^{3} - 1619 T^{4} + 180 p T^{5} + 37 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 10 p T^{5} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - 6 T + 13 T^{2} + 60 T^{3} - 659 T^{4} + 60 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 19 T^{2} + 120 T^{3} + 721 T^{4} + 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 834 p T^{5} + 113 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 13 T + 27 T^{2} - 445 T^{3} - 4264 T^{4} - 445 p T^{5} + 27 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 444 p T^{5} + 53 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2 T + 17 T^{2} + 130 T^{3} + 761 T^{4} + 130 p T^{5} + 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 21 T + 253 T^{2} - 2535 T^{3} + 21196 T^{4} - 2535 p T^{5} + 253 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 20 T + 181 T^{2} - 1600 T^{3} + 14601 T^{4} - 1600 p T^{5} + 181 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 424 p T^{5} + 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 18 T + 77 T^{2} + 30 T^{3} + 961 T^{4} + 30 p T^{5} + 77 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 28 T + 313 T^{2} - 2126 T^{3} + 14805 T^{4} - 2126 p T^{5} + 313 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 3 T^{2} - 980 T^{3} - 9259 T^{4} - 980 p T^{5} + 3 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 83 | $C_4\times C_2$ | \( 1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 5 T - 79 T^{2} - 5 p T^{3} + 5276 T^{4} - 5 p^{2} T^{5} - 79 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 18 T + 47 T^{2} - 1740 T^{3} - 27059 T^{4} - 1740 p T^{5} + 47 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.31319146605504610263299610059, −7.22275908129919476530836129504, −7.04094619460806537473878332181, −6.81476382275223761451325622468, −6.48248691457763743645028518979, −6.27951280289718200240354325684, −6.00854744204745203797476059356, −5.84128735457170805485733569146, −5.77752772553210582766396057707, −5.30454528423442238318683946454, −5.16440279518087057408399257984, −4.96928964668002958252492412578, −4.80977481514259225137322255427, −3.86944415383830531469308315959, −3.77673532502362526412645719431, −3.73619013067644831075704278417, −3.36975239480180264199684527892, −3.18758128842824047159457646974, −3.16045091177649475566511807503, −2.65698472824213349100530579633, −2.13809364458771241262066325792, −1.62484421106686713294202807402, −1.36972902705232675387277515565, −0.59181944214599629775821325079, −0.47202345352333535556760397766,
0.47202345352333535556760397766, 0.59181944214599629775821325079, 1.36972902705232675387277515565, 1.62484421106686713294202807402, 2.13809364458771241262066325792, 2.65698472824213349100530579633, 3.16045091177649475566511807503, 3.18758128842824047159457646974, 3.36975239480180264199684527892, 3.73619013067644831075704278417, 3.77673532502362526412645719431, 3.86944415383830531469308315959, 4.80977481514259225137322255427, 4.96928964668002958252492412578, 5.16440279518087057408399257984, 5.30454528423442238318683946454, 5.77752772553210582766396057707, 5.84128735457170805485733569146, 6.00854744204745203797476059356, 6.27951280289718200240354325684, 6.48248691457763743645028518979, 6.81476382275223761451325622468, 7.04094619460806537473878332181, 7.22275908129919476530836129504, 7.31319146605504610263299610059
