| L(s) = 1 | − 2.21·2-s − 3-s + 2.90·4-s − 1.68·5-s + 2.21·6-s + 7-s − 2·8-s + 9-s + 3.73·10-s + 5.80·11-s − 2.90·12-s + 1.09·13-s − 2.21·14-s + 1.68·15-s − 1.37·16-s + 17-s − 2.21·18-s + 1.90·19-s − 4.90·20-s − 21-s − 12.8·22-s − 1.83·23-s + 2·24-s − 2.14·25-s − 2.42·26-s − 27-s + 2.90·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.45·4-s − 0.755·5-s + 0.903·6-s + 0.377·7-s − 0.707·8-s + 0.333·9-s + 1.18·10-s + 1.75·11-s − 0.838·12-s + 0.304·13-s − 0.591·14-s + 0.436·15-s − 0.344·16-s + 0.242·17-s − 0.521·18-s + 0.436·19-s − 1.09·20-s − 0.218·21-s − 2.74·22-s − 0.382·23-s + 0.408·24-s − 0.429·25-s − 0.476·26-s − 0.192·27-s + 0.548·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 5.80T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 23 | \( 1 + 1.83T + 23T^{2} \) |
| 29 | \( 1 - 5.80T + 29T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 + 4.23T + 43T^{2} \) |
| 47 | \( 1 - 0.785T + 47T^{2} \) |
| 53 | \( 1 - 14.4T + 53T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 + 0.407T + 71T^{2} \) |
| 73 | \( 1 + 0.260T + 73T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 3.11T + 97T^{2} \) |
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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
−8.291546743712249470932338908687, −7.37175633377474571129799991821, −6.92810216999407434916131898409, −6.20201726466292473034902984876, −5.15557459913576389999642639889, −4.14637997021622905044651021971, −3.45699692394867264586168534729, −1.83099752204562142971899098118, −1.18828196683374123072640667184, 0,
1.18828196683374123072640667184, 1.83099752204562142971899098118, 3.45699692394867264586168534729, 4.14637997021622905044651021971, 5.15557459913576389999642639889, 6.20201726466292473034902984876, 6.92810216999407434916131898409, 7.37175633377474571129799991821, 8.291546743712249470932338908687
