L(s) = 1 | + 2-s + 4-s − 3.23·5-s − 7-s + 8-s − 3.23·10-s − 0.763·11-s + 1.23·13-s − 14-s + 16-s + 7.70·17-s + 19-s − 3.23·20-s − 0.763·22-s − 5.23·23-s + 5.47·25-s + 1.23·26-s − 28-s − 8.47·29-s − 2·31-s + 32-s + 7.70·34-s + 3.23·35-s − 10.4·37-s + 38-s − 3.23·40-s − 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.44·5-s − 0.377·7-s + 0.353·8-s − 1.02·10-s − 0.230·11-s + 0.342·13-s − 0.267·14-s + 0.250·16-s + 1.86·17-s + 0.229·19-s − 0.723·20-s − 0.162·22-s − 1.09·23-s + 1.09·25-s + 0.242·26-s − 0.188·28-s − 1.57·29-s − 0.359·31-s + 0.176·32-s + 1.32·34-s + 0.546·35-s − 1.72·37-s + 0.162·38-s − 0.511·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 7.70T + 17T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 + 8.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 - 0.291T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 0.763T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 4.76T + 97T^{2} \) |
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
−8.292608787526367761448417101977, −7.67767091126886013159040336558, −7.21796464055017913074093899188, −6.09772118717141205057242737834, −5.41088315588738257726309277293, −4.44850391368855207575750412546, −3.47003141905797618430276200780, −3.31109280574613546947275114947, −1.63640655895288500424174291013, 0,
1.63640655895288500424174291013, 3.31109280574613546947275114947, 3.47003141905797618430276200780, 4.44850391368855207575750412546, 5.41088315588738257726309277293, 6.09772118717141205057242737834, 7.21796464055017913074093899188, 7.67767091126886013159040336558, 8.292608787526367761448417101977