| L(s) = 1 | − 0.711·2-s + 3-s − 1.49·4-s − 5-s − 0.711·6-s − 4.85·7-s + 2.48·8-s + 9-s + 0.711·10-s − 3.28·11-s − 1.49·12-s − 1.86·13-s + 3.45·14-s − 15-s + 1.22·16-s + 1.26·17-s − 0.711·18-s + 4.69·19-s + 1.49·20-s − 4.85·21-s + 2.33·22-s − 2.57·23-s + 2.48·24-s + 25-s + 1.32·26-s + 27-s + 7.25·28-s + ⋯ |
| L(s) = 1 | − 0.502·2-s + 0.577·3-s − 0.747·4-s − 0.447·5-s − 0.290·6-s − 1.83·7-s + 0.878·8-s + 0.333·9-s + 0.224·10-s − 0.989·11-s − 0.431·12-s − 0.516·13-s + 0.923·14-s − 0.258·15-s + 0.305·16-s + 0.306·17-s − 0.167·18-s + 1.07·19-s + 0.334·20-s − 1.05·21-s + 0.497·22-s − 0.537·23-s + 0.507·24-s + 0.200·25-s + 0.259·26-s + 0.192·27-s + 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 + 0.711T + 2T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 + 3.28T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 4.69T + 19T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 0.989T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 + 2.62T + 59T^{2} \) |
| 61 | \( 1 - 9.39T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 3.43T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 + 4.91T + 97T^{2} \) |
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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
−8.038637828588121924545010439006, −7.13018331314611981211250508103, −6.54473253765704610553353828449, −5.51091300667265190472506800070, −4.76362913811006828744837596314, −3.90441309978470139193247906966, −3.13519013629177558458357156591, −2.60719470112486722928854687872, −0.968947578481608265822915899899, 0,
0.968947578481608265822915899899, 2.60719470112486722928854687872, 3.13519013629177558458357156591, 3.90441309978470139193247906966, 4.76362913811006828744837596314, 5.51091300667265190472506800070, 6.54473253765704610553353828449, 7.13018331314611981211250508103, 8.038637828588121924545010439006
