| L(s) = 1 | − 0.0822·2-s − 1.99·4-s − 4.31·5-s + 0.690·7-s + 0.328·8-s + 0.355·10-s + 2.95·11-s − 1.93·13-s − 0.0567·14-s + 3.95·16-s + 2.07·17-s + 3.74·19-s + 8.61·20-s − 0.243·22-s − 4.34·23-s + 13.6·25-s + 0.159·26-s − 1.37·28-s + 8.10·29-s − 2.80·31-s − 0.982·32-s − 0.170·34-s − 2.98·35-s − 9.72·37-s − 0.308·38-s − 1.41·40-s + 4.09·41-s + ⋯ |
| L(s) = 1 | − 0.0581·2-s − 0.996·4-s − 1.93·5-s + 0.261·7-s + 0.116·8-s + 0.112·10-s + 0.891·11-s − 0.536·13-s − 0.0151·14-s + 0.989·16-s + 0.503·17-s + 0.859·19-s + 1.92·20-s − 0.0518·22-s − 0.906·23-s + 2.73·25-s + 0.0312·26-s − 0.260·28-s + 1.50·29-s − 0.503·31-s − 0.173·32-s − 0.0292·34-s − 0.504·35-s − 1.59·37-s − 0.0499·38-s − 0.224·40-s + 0.639·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 0.0822T + 2T^{2} \) |
| 5 | \( 1 + 4.31T + 5T^{2} \) |
| 7 | \( 1 - 0.690T + 7T^{2} \) |
| 11 | \( 1 - 2.95T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 + 4.34T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 + 2.80T + 31T^{2} \) |
| 37 | \( 1 + 9.72T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 + 6.71T + 47T^{2} \) |
| 53 | \( 1 - 0.0484T + 53T^{2} \) |
| 59 | \( 1 + 4.50T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 + 0.964T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 - 16.4T + 73T^{2} \) |
| 79 | \( 1 + 6.83T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 - 2.36T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.588133868296134398727626766526, −7.909462627913316869593344141563, −7.46003007756804869945673824219, −6.46177592207931308093184589781, −5.15042662922262071258662270082, −4.55373164788228280555974272130, −3.79282232662482155275104563544, −3.17565358476256335423824917819, −1.19080940015765405079232093115, 0,
1.19080940015765405079232093115, 3.17565358476256335423824917819, 3.79282232662482155275104563544, 4.55373164788228280555974272130, 5.15042662922262071258662270082, 6.46177592207931308093184589781, 7.46003007756804869945673824219, 7.909462627913316869593344141563, 8.588133868296134398727626766526
