| L(s) = 1 | − 2-s + 4-s − 1.51·5-s − 2.59·7-s − 8-s + 1.51·10-s + 5.00·11-s − 4.82·13-s + 2.59·14-s + 16-s + 0.863·17-s − 7.00·19-s − 1.51·20-s − 5.00·22-s − 2.70·25-s + 4.82·26-s − 2.59·28-s − 3.11·29-s − 1.17·31-s − 32-s − 0.863·34-s + 3.92·35-s − 0.602·37-s + 7.00·38-s + 1.51·40-s − 5.29·41-s + 1.45·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.676·5-s − 0.980·7-s − 0.353·8-s + 0.478·10-s + 1.50·11-s − 1.33·13-s + 0.693·14-s + 0.250·16-s + 0.209·17-s − 1.60·19-s − 0.338·20-s − 1.06·22-s − 0.541·25-s + 0.945·26-s − 0.490·28-s − 0.578·29-s − 0.210·31-s − 0.176·32-s − 0.148·34-s + 0.663·35-s − 0.0989·37-s + 1.13·38-s + 0.239·40-s − 0.826·41-s + 0.222·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4704306759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4704306759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 5.00T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 0.863T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 0.602T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 1.45T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 4.23T + 53T^{2} \) |
| 59 | \( 1 + 3.98T + 59T^{2} \) |
| 61 | \( 1 + 7.26T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 5.88T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 2.64T + 83T^{2} \) |
| 89 | \( 1 - 5.23T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
| 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
−7.58324470711170513512140720006, −7.16388030792593431366367682105, −6.39757920659766763355097501306, −6.02065772182514300078145129095, −4.82044224197511044786900472132, −4.03332001735916290979713927927, −3.47953325721634153357432606155, −2.51061030000178738907618271973, −1.65786405919387265572134620499, −0.35572675913443251697285303426,
0.35572675913443251697285303426, 1.65786405919387265572134620499, 2.51061030000178738907618271973, 3.47953325721634153357432606155, 4.03332001735916290979713927927, 4.82044224197511044786900472132, 6.02065772182514300078145129095, 6.39757920659766763355097501306, 7.16388030792593431366367682105, 7.58324470711170513512140720006
