| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 5.23·11-s + 12-s + 4.85·13-s − 2·14-s + 16-s − 7.85·17-s + 18-s − 2.76·19-s − 2·21-s − 5.23·22-s − 6·23-s + 24-s + 4.85·26-s + 27-s − 2·28-s − 1.38·29-s + 3.70·31-s + 32-s − 5.23·33-s − 7.85·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 0.333·9-s − 1.57·11-s + 0.288·12-s + 1.34·13-s − 0.534·14-s + 0.250·16-s − 1.90·17-s + 0.235·18-s − 0.634·19-s − 0.436·21-s − 1.11·22-s − 1.25·23-s + 0.204·24-s + 0.951·26-s + 0.192·27-s − 0.377·28-s − 0.256·29-s + 0.666·31-s + 0.176·32-s − 0.911·33-s − 1.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + 4.76T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.259633647654173618686785698789, −7.36611392450256727299891951985, −6.39710669728980340164309265892, −6.12689135219086971231296965488, −4.97397316065943581066659257845, −4.23696750095191986829772141844, −3.44511602104617996566977121073, −2.62873430187901273371634989070, −1.87174870722852918343160449398, 0,
1.87174870722852918343160449398, 2.62873430187901273371634989070, 3.44511602104617996566977121073, 4.23696750095191986829772141844, 4.97397316065943581066659257845, 6.12689135219086971231296965488, 6.39710669728980340164309265892, 7.36611392450256727299891951985, 8.259633647654173618686785698789
