| L(s) = 1 | − 64·4-s − 3.19e3·9-s − 2.16e3·11-s + 4.09e3·16-s − 6.69e4·19-s − 2.50e5·29-s − 1.47e5·31-s + 2.04e5·36-s − 4.53e4·41-s + 1.38e5·44-s − 2.18e5·49-s − 2.19e6·59-s − 8.45e5·61-s − 2.62e5·64-s − 4.57e6·71-s + 4.28e6·76-s + 4.03e6·79-s + 5.42e6·81-s − 4.37e6·89-s + 6.92e6·99-s + 8.66e5·101-s − 3.20e7·109-s + 1.60e7·116-s − 3.54e7·121-s + 9.44e6·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.46·9-s − 0.490·11-s + 1/4·16-s − 2.23·19-s − 1.90·29-s − 0.889·31-s + 0.730·36-s − 0.102·41-s + 0.245·44-s − 0.265·49-s − 1.39·59-s − 0.477·61-s − 1/8·64-s − 1.51·71-s + 1.11·76-s + 0.921·79-s + 1.13·81-s − 0.657·89-s + 0.716·99-s + 0.0836·101-s − 2.37·109-s + 0.953·116-s − 1.81·121-s + 0.444·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 355 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 218870 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 1083 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 95598010 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 182154985 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 33485 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6775568650 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4320 p T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 73798 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 33106356790 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 22683 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 533607600310 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 298337578610 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2223481045750 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 1098360 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 422998 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 5575096711405 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2287428 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18513232750055 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2019250 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 9296355939035 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2185935 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 127681716222910 T^{2} + p^{14} T^{4} \) |
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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
−14.37608967902891957404635912309, −13.02720046326127208991362982573, −12.88123770649312800336407759976, −12.03922250895695304923172781398, −11.35896328845855456431427803407, −10.77384414669595965826543634727, −10.42354110184372121938524836929, −9.195241619677209107294753836989, −9.102756940554523753054050199878, −8.228167752248924171994145447757, −7.80295365569050377995688990454, −6.75684436994422189009079076497, −5.92674755316910020425576363470, −5.44803315489306644200442776995, −4.49325139267634280104137626393, −3.65652613527578717941828432631, −2.68779083382970985842253190328, −1.77821187710375870569819037595, 0, 0,
1.77821187710375870569819037595, 2.68779083382970985842253190328, 3.65652613527578717941828432631, 4.49325139267634280104137626393, 5.44803315489306644200442776995, 5.92674755316910020425576363470, 6.75684436994422189009079076497, 7.80295365569050377995688990454, 8.228167752248924171994145447757, 9.102756940554523753054050199878, 9.195241619677209107294753836989, 10.42354110184372121938524836929, 10.77384414669595965826543634727, 11.35896328845855456431427803407, 12.03922250895695304923172781398, 12.88123770649312800336407759976, 13.02720046326127208991362982573, 14.37608967902891957404635912309
