| L(s) = 1 | − 0.922·2-s − 3-s − 1.14·4-s − 0.0437·5-s + 0.922·6-s + 4.58·7-s + 2.90·8-s + 9-s + 0.0403·10-s + 2.66·11-s + 1.14·12-s − 0.617·13-s − 4.22·14-s + 0.0437·15-s − 0.381·16-s + 17-s − 0.922·18-s − 0.396·19-s + 0.0502·20-s − 4.58·21-s − 2.46·22-s − 0.881·23-s − 2.90·24-s − 4.99·25-s + 0.569·26-s − 27-s − 5.26·28-s + ⋯ |
| L(s) = 1 | − 0.652·2-s − 0.577·3-s − 0.574·4-s − 0.0195·5-s + 0.376·6-s + 1.73·7-s + 1.02·8-s + 0.333·9-s + 0.0127·10-s + 0.804·11-s + 0.331·12-s − 0.171·13-s − 1.12·14-s + 0.0112·15-s − 0.0952·16-s + 0.242·17-s − 0.217·18-s − 0.0910·19-s + 0.0112·20-s − 0.999·21-s − 0.524·22-s − 0.183·23-s − 0.592·24-s − 0.999·25-s + 0.111·26-s − 0.192·27-s − 0.995·28-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 | \( 1 + 0.922T + 2T^{2} \) |
| 5 | \( 1 + 0.0437T + 5T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 + 0.617T + 13T^{2} \) |
| 19 | \( 1 + 0.396T + 19T^{2} \) |
| 23 | \( 1 + 0.881T + 23T^{2} \) |
| 29 | \( 1 + 9.81T + 29T^{2} \) |
| 31 | \( 1 + 3.37T + 31T^{2} \) |
| 37 | \( 1 + 9.42T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 - 0.193T + 43T^{2} \) |
| 47 | \( 1 - 4.26T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.57T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 0.595T + 67T^{2} \) |
| 71 | \( 1 - 0.125T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 83 | \( 1 - 4.50T + 83T^{2} \) |
| 89 | \( 1 - 4.43T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−7.930865259949671015908598523475, −7.68747052609120818710966251231, −6.78579882854571552528017108046, −5.59368365790685045645295307027, −5.18143063857157794521805402867, −4.30628308182843814480510498074, −3.74098099481740454493864838016, −1.86401864906682501419700790699, −1.41566728249446923451179206850, 0,
1.41566728249446923451179206850, 1.86401864906682501419700790699, 3.74098099481740454493864838016, 4.30628308182843814480510498074, 5.18143063857157794521805402867, 5.59368365790685045645295307027, 6.78579882854571552528017108046, 7.68747052609120818710966251231, 7.930865259949671015908598523475
