| L(s) = 1 | − 2-s + 4-s + 3.23·5-s − 8-s − 3.23·10-s − 11-s − 1.23·13-s + 16-s − 6.47·17-s + 2.76·19-s + 3.23·20-s + 22-s − 4·23-s + 5.47·25-s + 1.23·26-s + 4.47·29-s − 2·31-s − 32-s + 6.47·34-s − 10.9·37-s − 2.76·38-s − 3.23·40-s + 6.47·41-s − 1.52·43-s − 44-s + 4·46-s − 2·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.353·8-s − 1.02·10-s − 0.301·11-s − 0.342·13-s + 0.250·16-s − 1.56·17-s + 0.634·19-s + 0.723·20-s + 0.213·22-s − 0.834·23-s + 1.09·25-s + 0.242·26-s + 0.830·29-s − 0.359·31-s − 0.176·32-s + 1.10·34-s − 1.79·37-s − 0.448·38-s − 0.511·40-s + 1.01·41-s − 0.232·43-s − 0.150·44-s + 0.589·46-s − 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 7.23T + 59T^{2} \) |
| 61 | \( 1 - 5.23T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
| 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
−7.27106900226214349429361205246, −6.67163897016025209483958688862, −6.11576293597130253520701997911, −5.39039573573711553871805384457, −4.77443602414202535048391254339, −3.69928742332344896828782178825, −2.58674712908158369260407786845, −2.16452390180340136583866657341, −1.32155024583528464959673079686, 0,
1.32155024583528464959673079686, 2.16452390180340136583866657341, 2.58674712908158369260407786845, 3.69928742332344896828782178825, 4.77443602414202535048391254339, 5.39039573573711553871805384457, 6.11576293597130253520701997911, 6.67163897016025209483958688862, 7.27106900226214349429361205246
