| L(s) = 1 | − 3.10·3-s + 5-s + 7-s + 6.62·9-s − 11-s − 3.62·13-s − 3.10·15-s − 4.20·17-s + 8.15·19-s − 3.10·21-s − 0.897·23-s + 25-s − 11.2·27-s − 7.30·29-s + 3.42·31-s + 3.10·33-s + 35-s − 1.10·37-s + 11.2·39-s + 12.3·41-s − 10.6·43-s + 6.62·45-s + 1.15·47-s + 49-s + 13.0·51-s + 11.3·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 0.447·5-s + 0.377·7-s + 2.20·9-s − 0.301·11-s − 1.00·13-s − 0.801·15-s − 1.01·17-s + 1.87·19-s − 0.677·21-s − 0.187·23-s + 0.200·25-s − 2.16·27-s − 1.35·29-s + 0.614·31-s + 0.540·33-s + 0.169·35-s − 0.181·37-s + 1.80·39-s + 1.92·41-s − 1.62·43-s + 0.987·45-s + 0.168·47-s + 0.142·49-s + 1.82·51-s + 1.55·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9083257990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9083257990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.10T + 3T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 - 8.15T + 19T^{2} \) |
| 23 | \( 1 + 0.897T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 1.15T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 7.25T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 - 8.41T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 - 0.205T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 2.20T + 83T^{2} \) |
| 89 | \( 1 + 7.45T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
| 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
−7.68234571362444453298513924062, −7.26765949487601721034436396335, −6.54955471571434654483315538366, −5.77760234751686823992690139057, −5.24706741901246797976673543355, −4.79629153123379106836435646031, −3.92373527899100765570343554066, −2.59884627036991048519611456286, −1.59951909010789060683837956891, −0.56185200369579379958390269150,
0.56185200369579379958390269150, 1.59951909010789060683837956891, 2.59884627036991048519611456286, 3.92373527899100765570343554066, 4.79629153123379106836435646031, 5.24706741901246797976673543355, 5.77760234751686823992690139057, 6.54955471571434654483315538366, 7.26765949487601721034436396335, 7.68234571362444453298513924062
