| L(s) = 1 | + 2.33·2-s − 0.00757·3-s + 3.46·4-s − 0.0177·6-s − 7-s + 3.41·8-s − 2.99·9-s − 4.99·11-s − 0.0262·12-s + 3.87·13-s − 2.33·14-s + 1.05·16-s + 0.475·17-s − 7.01·18-s − 1.54·19-s + 0.00757·21-s − 11.6·22-s − 23-s − 0.0258·24-s + 9.06·26-s + 0.0454·27-s − 3.46·28-s − 9.38·29-s − 11.0·31-s − 4.35·32-s + 0.0378·33-s + 1.11·34-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 0.00437·3-s + 1.73·4-s − 0.00722·6-s − 0.377·7-s + 1.20·8-s − 0.999·9-s − 1.50·11-s − 0.00756·12-s + 1.07·13-s − 0.624·14-s + 0.264·16-s + 0.115·17-s − 1.65·18-s − 0.353·19-s + 0.00165·21-s − 2.48·22-s − 0.208·23-s − 0.00528·24-s + 1.77·26-s + 0.00874·27-s − 0.654·28-s − 1.74·29-s − 1.99·31-s − 0.770·32-s + 0.00658·33-s + 0.190·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 2.33T + 2T^{2} \) |
| 3 | \( 1 + 0.00757T + 3T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 - 0.475T + 17T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 29 | \( 1 + 9.38T + 29T^{2} \) |
| 31 | \( 1 + 11.0T + 31T^{2} \) |
| 37 | \( 1 - 4.42T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 3.40T + 43T^{2} \) |
| 47 | \( 1 - 8.07T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 8.24T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 6.08T + 89T^{2} \) |
| 97 | \( 1 + 3.91T + 97T^{2} \) |
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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
−7.84413312949056977962917307322, −7.18951124590314177070416937922, −6.17610117086979798874073283130, −5.64396657560904164870036704759, −5.30947636772553993894771907313, −4.17035774942377285517625280944, −3.51017651085911852444850799906, −2.79854853166813908159313942533, −1.98522185258553126732745089697, 0,
1.98522185258553126732745089697, 2.79854853166813908159313942533, 3.51017651085911852444850799906, 4.17035774942377285517625280944, 5.30947636772553993894771907313, 5.64396657560904164870036704759, 6.17610117086979798874073283130, 7.18951124590314177070416937922, 7.84413312949056977962917307322
