| L(s) = 1 | + 2.38·2-s − 8.62·3-s − 2.33·4-s + 6.85·5-s − 20.5·6-s − 24.5·8-s + 47.3·9-s + 16.3·10-s + 65.2·11-s + 20.1·12-s − 39.2·13-s − 59.1·15-s − 39.8·16-s − 14.0·17-s + 112.·18-s − 99.3·19-s − 16.0·20-s + 155.·22-s + 7.81·23-s + 212.·24-s − 77.9·25-s − 93.4·26-s − 175.·27-s + 107.·29-s − 140.·30-s − 21.1·31-s + 101.·32-s + ⋯ |
| L(s) = 1 | + 0.841·2-s − 1.65·3-s − 0.291·4-s + 0.613·5-s − 1.39·6-s − 1.08·8-s + 1.75·9-s + 0.516·10-s + 1.78·11-s + 0.484·12-s − 0.837·13-s − 1.01·15-s − 0.623·16-s − 0.200·17-s + 1.47·18-s − 1.19·19-s − 0.178·20-s + 1.50·22-s + 0.0708·23-s + 1.80·24-s − 0.623·25-s − 0.705·26-s − 1.25·27-s + 0.690·29-s − 0.856·30-s − 0.122·31-s + 0.562·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 - 2.38T + 8T^{2} \) |
| 3 | \( 1 + 8.62T + 27T^{2} \) |
| 5 | \( 1 - 6.85T + 125T^{2} \) |
| 11 | \( 1 - 65.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.81T + 1.21e4T^{2} \) |
| 29 | \( 1 - 107.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 21.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 87.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 416.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 217.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 76.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 219.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 194.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 314.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 55.2T + 5.71e5T^{2} \) |
| 89 | \( 1 - 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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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
−8.322445085669345605761024051444, −6.82494032526070208912059572737, −6.53247412703173130910367159997, −5.81543027787239577586741741442, −5.15610641719123050398147849519, −4.38065101286243547086502511270, −3.82994096460425697170007122122, −2.26913488204921101140426535835, −1.04734588847442627582764891229, 0,
1.04734588847442627582764891229, 2.26913488204921101140426535835, 3.82994096460425697170007122122, 4.38065101286243547086502511270, 5.15610641719123050398147849519, 5.81543027787239577586741741442, 6.53247412703173130910367159997, 6.82494032526070208912059572737, 8.322445085669345605761024051444
