| L(s) = 1 | + 2-s + 2.68·3-s + 4-s − 2.52·5-s + 2.68·6-s + 7-s + 8-s + 4.21·9-s − 2.52·10-s − 3.41·11-s + 2.68·12-s − 0.744·13-s + 14-s − 6.78·15-s + 16-s + 0.526·17-s + 4.21·18-s − 7.37·19-s − 2.52·20-s + 2.68·21-s − 3.41·22-s + 2.68·24-s + 1.38·25-s − 0.744·26-s + 3.25·27-s + 28-s − 4.92·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.55·3-s + 0.5·4-s − 1.12·5-s + 1.09·6-s + 0.377·7-s + 0.353·8-s + 1.40·9-s − 0.798·10-s − 1.02·11-s + 0.775·12-s − 0.206·13-s + 0.267·14-s − 1.75·15-s + 0.250·16-s + 0.127·17-s + 0.992·18-s − 1.69·19-s − 0.564·20-s + 0.586·21-s − 0.727·22-s + 0.548·24-s + 0.276·25-s − 0.146·26-s + 0.626·27-s + 0.188·28-s − 0.914·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 + 2.52T + 5T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 + 0.744T + 13T^{2} \) |
| 17 | \( 1 - 0.526T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 + 2.18T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 5.68T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 6.27T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 0.603T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 0.631T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
| 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
−7.69202782070061037023935908672, −7.17078929684890732644516994716, −6.19357425751600044716099472268, −5.16916285373074580343731506948, −4.49585540475849936862785230356, −3.76703877949356341181853191954, −3.33585979290668542498784171864, −2.38213332186183694826024832539, −1.83730996047445307291389016050, 0,
1.83730996047445307291389016050, 2.38213332186183694826024832539, 3.33585979290668542498784171864, 3.76703877949356341181853191954, 4.49585540475849936862785230356, 5.16916285373074580343731506948, 6.19357425751600044716099472268, 7.17078929684890732644516994716, 7.69202782070061037023935908672
