| L(s) = 1 | + 2-s − 2.33·3-s + 4-s − 2.33·6-s + 3.77·7-s + 8-s + 2.44·9-s − 3.77·11-s − 2.33·12-s + 3.17·13-s + 3.77·14-s + 16-s − 1.39·17-s + 2.44·18-s + 3.91·19-s − 8.80·21-s − 3.77·22-s + 0.891·23-s − 2.33·24-s + 3.17·26-s + 1.30·27-s + 3.77·28-s + 0.0492·29-s − 5.58·31-s + 32-s + 8.80·33-s − 1.39·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·3-s + 0.5·4-s − 0.952·6-s + 1.42·7-s + 0.353·8-s + 0.813·9-s − 1.13·11-s − 0.673·12-s + 0.880·13-s + 1.00·14-s + 0.250·16-s − 0.337·17-s + 0.575·18-s + 0.897·19-s − 1.92·21-s − 0.804·22-s + 0.185·23-s − 0.476·24-s + 0.622·26-s + 0.250·27-s + 0.713·28-s + 0.00913·29-s − 1.00·31-s + 0.176·32-s + 1.53·33-s − 0.238·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.873117746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.873117746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.33T + 3T^{2} \) |
| 7 | \( 1 - 3.77T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 - 3.17T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 - 0.891T + 23T^{2} \) |
| 29 | \( 1 - 0.0492T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - 7.04T + 37T^{2} \) |
| 41 | \( 1 + 1.48T + 41T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 0.690T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 - 5.16T + 73T^{2} \) |
| 79 | \( 1 + 1.80T + 79T^{2} \) |
| 83 | \( 1 + 9.96T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 0.0901T + 97T^{2} \) |
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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
−10.11786907915969445080972221245, −8.741734996155812675289072868262, −7.85253265829644812949957613001, −7.11215120045000757755652806049, −6.02295408932589727708438262143, −5.35600553088722848839698008082, −4.91085236311740370166266310637, −3.87420591515630591006507556629, −2.38402247462264952478873044434, −1.03118147548760411110470121803,
1.03118147548760411110470121803, 2.38402247462264952478873044434, 3.87420591515630591006507556629, 4.91085236311740370166266310637, 5.35600553088722848839698008082, 6.02295408932589727708438262143, 7.11215120045000757755652806049, 7.85253265829644812949957613001, 8.741734996155812675289072868262, 10.11786907915969445080972221245
