| L(s) = 1 | − 1.41·2-s + 1.93·3-s − 2.73·6-s − 2.44·7-s + 2.82·8-s + 0.732·9-s − 1.26·11-s − 2.44·13-s + 3.46·14-s − 4.00·16-s + 4.38·17-s − 1.03·18-s − 7.19·19-s − 4.73·21-s + 1.79·22-s + 1.41·23-s + 5.46·24-s + 3.46·26-s − 4.38·27-s − 8.19·29-s + 31-s − 2.44·33-s − 6.19·34-s + 3.34·37-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 1.11·3-s − 1.11·6-s − 0.925·7-s + 0.999·8-s + 0.244·9-s − 0.382·11-s − 0.679·13-s + 0.925·14-s − 1.00·16-s + 1.06·17-s − 0.244·18-s − 1.65·19-s − 1.03·21-s + 0.382·22-s + 0.294·23-s + 1.11·24-s + 0.679·26-s − 0.843·27-s − 1.52·29-s + 0.179·31-s − 0.426·33-s − 1.06·34-s + 0.550·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 - 1.93T + 3T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 0.240T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 8.62T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 6.39T + 61T^{2} \) |
| 67 | \( 1 - 2.44T + 67T^{2} \) |
| 71 | \( 1 + 4.26T + 71T^{2} \) |
| 73 | \( 1 + 4.00T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 + 7.48T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−9.738449377551438211782823646977, −9.037912974242571750235977729560, −8.310988764294481681447721600466, −7.66820974083290026821829768080, −6.76301366390751630856057695998, −5.44020026897793139706150131997, −4.11745765140408628074048959264, −3.08531499884904106192682956613, −1.95744748745651875072696835761, 0,
1.95744748745651875072696835761, 3.08531499884904106192682956613, 4.11745765140408628074048959264, 5.44020026897793139706150131997, 6.76301366390751630856057695998, 7.66820974083290026821829768080, 8.310988764294481681447721600466, 9.037912974242571750235977729560, 9.738449377551438211782823646977
