| L(s) = 1 | + 4·2-s − 2·3-s + 8·4-s − 8·6-s − 6·7-s − 23·9-s + 32·11-s − 16·12-s + 38·13-s − 24·14-s − 64·16-s − 26·17-s − 92·18-s + 100·19-s + 12·21-s + 128·22-s + 78·23-s + 152·26-s + 100·27-s − 48·28-s − 50·29-s − 108·31-s − 256·32-s − 64·33-s − 104·34-s − 184·36-s − 266·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.384·3-s + 4-s − 0.544·6-s − 0.323·7-s − 0.851·9-s + 0.877·11-s − 0.384·12-s + 0.810·13-s − 0.458·14-s − 16-s − 0.370·17-s − 1.20·18-s + 1.20·19-s + 0.124·21-s + 1.24·22-s + 0.707·23-s + 1.14·26-s + 0.712·27-s − 0.323·28-s − 0.320·29-s − 0.625·31-s − 1.41·32-s − 0.337·33-s − 0.524·34-s − 0.851·36-s − 1.18·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.841899855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.841899855\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 26 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 + 50 T + p^{3} T^{2} \) |
| 31 | \( 1 + 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 442 T + p^{3} T^{2} \) |
| 47 | \( 1 - 514 T + p^{3} T^{2} \) |
| 53 | \( 1 + 2 T + p^{3} T^{2} \) |
| 59 | \( 1 - 500 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 126 T + p^{3} T^{2} \) |
| 71 | \( 1 - 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 878 T + p^{3} T^{2} \) |
| 79 | \( 1 - 600 T + p^{3} T^{2} \) |
| 83 | \( 1 + 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 150 T + p^{3} T^{2} \) |
| 97 | \( 1 + 386 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.84616040582696139050927983327, −15.60014711112066009658969979317, −14.35568464100147557129641128735, −13.43874505261327947885981935722, −12.09989993250832259186310102702, −11.17221725572966404795949135327, −9.022824012314923508899261884816, −6.60238089212847252374433704868, −5.34267571116045261373041245421, −3.47725905250379747646841008283,
3.47725905250379747646841008283, 5.34267571116045261373041245421, 6.60238089212847252374433704868, 9.022824012314923508899261884816, 11.17221725572966404795949135327, 12.09989993250832259186310102702, 13.43874505261327947885981935722, 14.35568464100147557129641128735, 15.60014711112066009658969979317, 16.84616040582696139050927983327
