| L(s) = 1 | + 1.45·2-s + 2.86·3-s + 0.121·4-s − 5-s + 4.16·6-s − 0.762·7-s − 2.73·8-s + 5.17·9-s − 1.45·10-s − 11-s + 0.347·12-s − 3.14·13-s − 1.11·14-s − 2.86·15-s − 4.22·16-s − 5.35·17-s + 7.54·18-s − 3.77·19-s − 0.121·20-s − 2.18·21-s − 1.45·22-s − 7.47·23-s − 7.82·24-s + 25-s − 4.57·26-s + 6.23·27-s − 0.0927·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 1.65·3-s + 0.0607·4-s − 0.447·5-s + 1.70·6-s − 0.288·7-s − 0.967·8-s + 1.72·9-s − 0.460·10-s − 0.301·11-s + 0.100·12-s − 0.872·13-s − 0.296·14-s − 0.738·15-s − 1.05·16-s − 1.29·17-s + 1.77·18-s − 0.865·19-s − 0.0271·20-s − 0.476·21-s − 0.310·22-s − 1.55·23-s − 1.59·24-s + 0.200·25-s − 0.898·26-s + 1.19·27-s − 0.0175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 3 | \( 1 - 2.86T + 3T^{2} \) |
| 7 | \( 1 + 0.762T + 7T^{2} \) |
| 13 | \( 1 + 3.14T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 + 3.77T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 - 0.158T + 37T^{2} \) |
| 41 | \( 1 - 9.35T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 - 5.11T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 + 8.64T + 67T^{2} \) |
| 71 | \( 1 - 5.75T + 71T^{2} \) |
| 79 | \( 1 - 1.29T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 1.92T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−8.235589355600869666267770619537, −7.38564078860139429605943024693, −6.65618592694321166800591821213, −5.78400075181824451077407691026, −4.56227738873806432952404812194, −4.27870332228873403667079857402, −3.47847244551738973157172909710, −2.64774484448802634776483914228, −2.12344983969183513796792149745, 0,
2.12344983969183513796792149745, 2.64774484448802634776483914228, 3.47847244551738973157172909710, 4.27870332228873403667079857402, 4.56227738873806432952404812194, 5.78400075181824451077407691026, 6.65618592694321166800591821213, 7.38564078860139429605943024693, 8.235589355600869666267770619537
