| L(s) = 1 | − 131·2-s + 3.50e3·4-s + 3.12e4·5-s + 4.96e5·7-s + 2.55e5·8-s − 4.09e6·10-s − 6.24e6·11-s + 1.76e6·13-s − 6.50e7·14-s − 7.91e6·16-s − 4.81e7·17-s + 3.84e7·19-s + 1.09e8·20-s + 8.18e8·22-s − 7.34e8·23-s + 7.32e8·25-s − 2.30e8·26-s + 1.74e9·28-s − 1.12e9·29-s + 6.10e9·31-s + 2.57e9·32-s + 6.30e9·34-s + 1.55e10·35-s − 4.68e10·37-s − 5.04e9·38-s + 7.98e9·40-s + 2.96e10·41-s + ⋯ |
| L(s) = 1 | − 1.44·2-s + 0.428·4-s + 0.894·5-s + 1.59·7-s + 0.344·8-s − 1.29·10-s − 1.06·11-s + 0.101·13-s − 2.30·14-s − 0.117·16-s − 0.483·17-s + 0.187·19-s + 0.383·20-s + 1.53·22-s − 1.03·23-s + 3/5·25-s − 0.146·26-s + 0.682·28-s − 0.351·29-s + 1.23·31-s + 0.423·32-s + 0.700·34-s + 1.42·35-s − 2.99·37-s − 0.271·38-s + 0.308·40-s + 0.973·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+13/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{6} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 131 T + 3413 p^{2} T^{2} + 131 p^{13} T^{3} + p^{26} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 10128 p^{2} T + 4833248606 p^{2} T^{2} - 10128 p^{15} T^{3} + p^{26} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 567808 p T + 577871634854 p^{2} T^{2} + 567808 p^{14} T^{3} + p^{26} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1761164 T + 606461761659534 T^{2} - 1761164 p^{13} T^{3} + p^{26} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 48151604 T + 17780958841745078 T^{2} + 48151604 p^{13} T^{3} + p^{26} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 38475344 T - 37450960045515402 T^{2} - 38475344 p^{13} T^{3} + p^{26} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 734264616 T + 1132235031622772206 T^{2} + 734264616 p^{13} T^{3} + p^{26} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1126474708 T - 2318209037061176962 T^{2} + 1126474708 p^{13} T^{3} + p^{26} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6100104312 T + 36788978328798625982 T^{2} - 6100104312 p^{13} T^{3} + p^{26} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 46819674548 T + \)\(10\!\cdots\!74\)\( T^{2} + 46819674548 p^{13} T^{3} + p^{26} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 29607909788 T + \)\(20\!\cdots\!22\)\( T^{2} - 29607909788 p^{13} T^{3} + p^{26} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 63189768520 T + \)\(30\!\cdots\!10\)\( T^{2} + 63189768520 p^{13} T^{3} + p^{26} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 105004415960 T + \)\(11\!\cdots\!10\)\( T^{2} + 105004415960 p^{13} T^{3} + p^{26} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 48425833796 T + \)\(10\!\cdots\!26\)\( T^{2} + 48425833796 p^{13} T^{3} + p^{26} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 722768180096 T + \)\(33\!\cdots\!58\)\( T^{2} + 722768180096 p^{13} T^{3} + p^{26} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 229984402180 T + \)\(25\!\cdots\!38\)\( T^{2} + 229984402180 p^{13} T^{3} + p^{26} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 359062188984 T + \)\(11\!\cdots\!38\)\( T^{2} - 359062188984 p^{13} T^{3} + p^{26} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 736545668224 T + \)\(18\!\cdots\!66\)\( T^{2} + 736545668224 p^{13} T^{3} + p^{26} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1494470740204 T + \)\(35\!\cdots\!46\)\( T^{2} + 1494470740204 p^{13} T^{3} + p^{26} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3526348590120 T + \)\(74\!\cdots\!78\)\( T^{2} - 3526348590120 p^{13} T^{3} + p^{26} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5631050986968 T + \)\(21\!\cdots\!58\)\( T^{2} + 5631050986968 p^{13} T^{3} + p^{26} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5913121459764 T + \)\(37\!\cdots\!18\)\( T^{2} + 5913121459764 p^{13} T^{3} + p^{26} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16383233322244 T + \)\(16\!\cdots\!38\)\( T^{2} - 16383233322244 p^{13} T^{3} + p^{26} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63604274484398746200366470408, −11.96966431872176344558457409124, −11.28163671797462387408966812252, −10.70231604839450367505876725035, −10.11316360654081564868371575209, −9.816885976573304768540598273605, −8.931009158294423181511878153388, −8.524746910734751247905055338606, −8.039472356997977014752522369470, −7.55044658002135311243096631444, −6.55247109106124805747332622355, −5.80195994313786696015143943662, −4.85418413571621809445417763957, −4.76556031076006729655015843376, −3.35179109022330814903077356246, −2.40429526860341645118354320284, −1.55214475298217895583316704616, −1.41924993575439483541311374456, 0, 0,
1.41924993575439483541311374456, 1.55214475298217895583316704616, 2.40429526860341645118354320284, 3.35179109022330814903077356246, 4.76556031076006729655015843376, 4.85418413571621809445417763957, 5.80195994313786696015143943662, 6.55247109106124805747332622355, 7.55044658002135311243096631444, 8.039472356997977014752522369470, 8.524746910734751247905055338606, 8.931009158294423181511878153388, 9.816885976573304768540598273605, 10.11316360654081564868371575209, 10.70231604839450367505876725035, 11.28163671797462387408966812252, 11.96966431872176344558457409124, 12.63604274484398746200366470408
