| L(s) = 1 | − 1.19·3-s + 4.07·5-s − 3.61·7-s − 1.56·9-s + 11-s − 1.47·13-s − 4.87·15-s − 3.27·17-s − 19-s + 4.32·21-s + 7.45·23-s + 11.6·25-s + 5.46·27-s + 1.02·29-s − 1.64·31-s − 1.19·33-s − 14.7·35-s − 6.71·37-s + 1.76·39-s − 3.92·41-s − 5.38·43-s − 6.39·45-s + 3.71·47-s + 6.09·49-s + 3.91·51-s − 0.102·53-s + 4.07·55-s + ⋯ |
| L(s) = 1 | − 0.690·3-s + 1.82·5-s − 1.36·7-s − 0.523·9-s + 0.301·11-s − 0.410·13-s − 1.25·15-s − 0.793·17-s − 0.229·19-s + 0.944·21-s + 1.55·23-s + 2.32·25-s + 1.05·27-s + 0.190·29-s − 0.296·31-s − 0.208·33-s − 2.49·35-s − 1.10·37-s + 0.283·39-s − 0.612·41-s − 0.820·43-s − 0.953·45-s + 0.542·47-s + 0.870·49-s + 0.547·51-s − 0.0141·53-s + 0.549·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 3.71T + 47T^{2} \) |
| 53 | \( 1 + 0.102T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
| 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
−8.602959830519736450429035103709, −6.98851379764319489036766953933, −6.65670511448495770766258014692, −6.01617336471164392546328907233, −5.40382704741309113114334322312, −4.68603211203720698716993593959, −3.20717349582467339245621173779, −2.60896616369521154740192171692, −1.46153496748798818328200574360, 0,
1.46153496748798818328200574360, 2.60896616369521154740192171692, 3.20717349582467339245621173779, 4.68603211203720698716993593959, 5.40382704741309113114334322312, 6.01617336471164392546328907233, 6.65670511448495770766258014692, 6.98851379764319489036766953933, 8.602959830519736450429035103709
