| L(s) = 1 | − 3.05·2-s + 3·3-s + 1.32·4-s − 6.59·5-s − 9.15·6-s − 7.71·7-s + 20.3·8-s + 9·9-s + 20.1·10-s + 11·11-s + 3.96·12-s − 86.2·13-s + 23.5·14-s − 19.7·15-s − 72.8·16-s + 11.7·17-s − 27.4·18-s + 73.1·19-s − 8.72·20-s − 23.1·21-s − 33.5·22-s − 9.70·23-s + 61.1·24-s − 81.4·25-s + 263.·26-s + 27·27-s − 10.2·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.165·4-s − 0.590·5-s − 0.623·6-s − 0.416·7-s + 0.901·8-s + 0.333·9-s + 0.636·10-s + 0.301·11-s + 0.0954·12-s − 1.84·13-s + 0.449·14-s − 0.340·15-s − 1.13·16-s + 0.167·17-s − 0.359·18-s + 0.883·19-s − 0.0975·20-s − 0.240·21-s − 0.325·22-s − 0.0879·23-s + 0.520·24-s − 0.651·25-s + 1.98·26-s + 0.192·27-s − 0.0688·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 3.05T + 8T^{2} \) |
| 5 | \( 1 + 6.59T + 125T^{2} \) |
| 7 | \( 1 + 7.71T + 343T^{2} \) |
| 13 | \( 1 + 86.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 11.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.70T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 343.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 334.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 151.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 541.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 787.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 107.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 281.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 126.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 460.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.45e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 591.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.371494785962746062746635029074, −7.71024011906675179767968007813, −7.33357277474714220107795529429, −6.33869955037103738929845019753, −4.94108257035178289793337390725, −4.33029543813708507455769406607, −3.19550485546080572877485471195, −2.26607541963967406460830724621, −0.997738334088170008427834181563, 0,
0.997738334088170008427834181563, 2.26607541963967406460830724621, 3.19550485546080572877485471195, 4.33029543813708507455769406607, 4.94108257035178289793337390725, 6.33869955037103738929845019753, 7.33357277474714220107795529429, 7.71024011906675179767968007813, 8.371494785962746062746635029074
