L(s) = 1 | + 1.48·2-s + 0.193·4-s − 5-s + 1.19·7-s − 2.67·8-s − 1.48·10-s − 4.15·11-s + 2.96·13-s + 1.76·14-s − 4.35·16-s − 5.50·17-s − 3.19·19-s − 0.193·20-s − 6.15·22-s − 1.84·23-s + 25-s + 4.38·26-s + 0.231·28-s + 29-s − 4.80·31-s − 1.09·32-s − 8.15·34-s − 1.19·35-s − 9.50·37-s − 4.73·38-s + 2.67·40-s + 11.2·41-s + ⋯ |
L(s) = 1 | + 1.04·2-s + 0.0969·4-s − 0.447·5-s + 0.451·7-s − 0.945·8-s − 0.468·10-s − 1.25·11-s + 0.821·13-s + 0.472·14-s − 1.08·16-s − 1.33·17-s − 0.732·19-s − 0.0433·20-s − 1.31·22-s − 0.384·23-s + 0.200·25-s + 0.860·26-s + 0.0437·28-s + 0.185·29-s − 0.863·31-s − 0.193·32-s − 1.39·34-s − 0.201·35-s − 1.56·37-s − 0.767·38-s + 0.422·40-s + 1.76·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.15T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 5.50T + 17T^{2} \) |
| 19 | \( 1 + 3.19T + 19T^{2} \) |
| 23 | \( 1 + 1.84T + 23T^{2} \) |
| 31 | \( 1 + 4.80T + 31T^{2} \) |
| 37 | \( 1 + 9.50T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 0.0303T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 1.35T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 - 8.88T + 61T^{2} \) |
| 67 | \( 1 - 5.84T + 67T^{2} \) |
| 71 | \( 1 - 1.27T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 4.93T + 79T^{2} \) |
| 83 | \( 1 + 4.41T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 1.38T + 97T^{2} \) |
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
−8.985366969508367006331166122664, −8.499775554627435530103670311208, −7.58528448555998200914355214806, −6.53196820676728138265894050184, −5.71773564171144200719753467143, −4.83419416263156387978111592624, −4.19989824034651920284450046856, −3.23904655753008815214552442450, −2.12298168780212601923935844999, 0,
2.12298168780212601923935844999, 3.23904655753008815214552442450, 4.19989824034651920284450046856, 4.83419416263156387978111592624, 5.71773564171144200719753467143, 6.53196820676728138265894050184, 7.58528448555998200914355214806, 8.499775554627435530103670311208, 8.985366969508367006331166122664