| L(s) = 1 | + 0.287·2-s + 3-s − 1.91·4-s + 0.287·6-s − 1.12·8-s + 9-s − 3.33·11-s − 1.91·12-s + 4.54·13-s + 3.51·16-s − 5.54·17-s + 0.287·18-s − 1.65·19-s − 0.956·22-s + 7.63·23-s − 1.12·24-s + 1.30·26-s + 27-s − 0.118·29-s − 6.26·31-s + 3.26·32-s − 3.33·33-s − 1.59·34-s − 1.91·36-s − 7.75·37-s − 0.476·38-s + 4.54·39-s + ⋯ |
| L(s) = 1 | + 0.203·2-s + 0.577·3-s − 0.958·4-s + 0.117·6-s − 0.397·8-s + 0.333·9-s − 1.00·11-s − 0.553·12-s + 1.26·13-s + 0.877·16-s − 1.34·17-s + 0.0677·18-s − 0.380·19-s − 0.204·22-s + 1.59·23-s − 0.229·24-s + 0.256·26-s + 0.192·27-s − 0.0220·29-s − 1.12·31-s + 0.576·32-s − 0.579·33-s − 0.273·34-s − 0.319·36-s − 1.27·37-s − 0.0772·38-s + 0.728·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.287T + 2T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 7.63T + 23T^{2} \) |
| 29 | \( 1 + 0.118T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 + 7.75T + 37T^{2} \) |
| 41 | \( 1 - 0.0701T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 - 6.38T + 47T^{2} \) |
| 53 | \( 1 + 0.739T + 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 + 3.03T + 67T^{2} \) |
| 71 | \( 1 + 3.77T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 1.22T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 3.04T + 97T^{2} \) |
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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
−8.420003832376549825465369259455, −7.52169979694580295198376094278, −6.73099924919233677991081947688, −5.76838716268651223823080828739, −5.03383098039058290785453727663, −4.28394530354246107915254612402, −3.50623839435450937314198996412, −2.70534990639154177324241042954, −1.46504004314762340670932972746, 0,
1.46504004314762340670932972746, 2.70534990639154177324241042954, 3.50623839435450937314198996412, 4.28394530354246107915254612402, 5.03383098039058290785453727663, 5.76838716268651223823080828739, 6.73099924919233677991081947688, 7.52169979694580295198376094278, 8.420003832376549825465369259455
