| L(s) = 1 | + 3·4-s + 5·9-s − 8·11-s + 5·16-s + 12·19-s − 14·31-s + 15·36-s − 4·41-s − 24·44-s + 13·49-s − 16·59-s + 20·61-s + 3·64-s + 14·71-s + 36·76-s + 22·79-s + 16·81-s + 12·89-s − 40·99-s + 30·101-s + 8·109-s + 26·121-s − 42·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 5/3·9-s − 2.41·11-s + 5/4·16-s + 2.75·19-s − 2.51·31-s + 5/2·36-s − 0.624·41-s − 3.61·44-s + 13/7·49-s − 2.08·59-s + 2.56·61-s + 3/8·64-s + 1.66·71-s + 4.12·76-s + 2.47·79-s + 16/9·81-s + 1.27·89-s − 4.02·99-s + 2.98·101-s + 0.766·109-s + 2.36·121-s − 3.77·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582664386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582664386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−11.32747316071937976220647828529, −10.86272911899956796142579682521, −10.53977168770839914671963012133, −10.22430697620942944638591880988, −9.699677461183520456472066567750, −9.375267854060322023187195627850, −8.630823289347306924389441892267, −7.70435848304743941488685137839, −7.60084711130629951997117050429, −7.42938471590874538525234107269, −6.96919964975090527367086605617, −6.28447827990415011460858667792, −5.62190487616811240677112573918, −5.05764505512543381811566969493, −5.01455878607772551199993160919, −3.55529778655264588943773253420, −3.52247902799370925485161591156, −2.44349194376163007043486725423, −2.11470305054087019360518800532, −1.10409486004601865020106532370,
1.10409486004601865020106532370, 2.11470305054087019360518800532, 2.44349194376163007043486725423, 3.52247902799370925485161591156, 3.55529778655264588943773253420, 5.01455878607772551199993160919, 5.05764505512543381811566969493, 5.62190487616811240677112573918, 6.28447827990415011460858667792, 6.96919964975090527367086605617, 7.42938471590874538525234107269, 7.60084711130629951997117050429, 7.70435848304743941488685137839, 8.630823289347306924389441892267, 9.375267854060322023187195627850, 9.699677461183520456472066567750, 10.22430697620942944638591880988, 10.53977168770839914671963012133, 10.86272911899956796142579682521, 11.32747316071937976220647828529
