| L(s) = 1 | + 2.22·2-s − 3-s + 2.96·4-s − 2.22·6-s + 2.15·8-s + 9-s + 1.77·11-s − 2.96·12-s − 4.19·13-s − 1.13·16-s + 0.322·17-s + 2.22·18-s − 7.66·19-s + 3.96·22-s − 4.97·23-s − 2.15·24-s − 9.34·26-s − 27-s − 8.08·29-s − 4.67·31-s − 6.83·32-s − 1.77·33-s + 0.719·34-s + 2.96·36-s + 10.2·37-s − 17.0·38-s + 4.19·39-s + ⋯ |
| L(s) = 1 | + 1.57·2-s − 0.577·3-s + 1.48·4-s − 0.909·6-s + 0.760·8-s + 0.333·9-s + 0.536·11-s − 0.856·12-s − 1.16·13-s − 0.284·16-s + 0.0783·17-s + 0.525·18-s − 1.75·19-s + 0.845·22-s − 1.03·23-s − 0.439·24-s − 1.83·26-s − 0.192·27-s − 1.50·29-s − 0.838·31-s − 1.20·32-s − 0.309·33-s + 0.123·34-s + 0.494·36-s + 1.69·37-s − 2.77·38-s + 0.671·39-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 | \( 1 - 2.22T + 2T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 13 | \( 1 + 4.19T + 13T^{2} \) |
| 17 | \( 1 - 0.322T + 17T^{2} \) |
| 19 | \( 1 + 7.66T + 19T^{2} \) |
| 23 | \( 1 + 4.97T + 23T^{2} \) |
| 29 | \( 1 + 8.08T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 1.69T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 5.84T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 5.60T + 67T^{2} \) |
| 71 | \( 1 - 6.72T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 + 3.23T + 79T^{2} \) |
| 83 | \( 1 - 4.52T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 5.27T + 97T^{2} \) |
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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
−7.77392730988486122299209538619, −7.19964575265856971764462979398, −6.22115223084552657718728570847, −5.95827570516799109585872795691, −5.05664007916021817510124073443, −4.26394353215895716922063400668, −3.90673928877008595355081213073, −2.63241719874562319162363862139, −1.88551191967626174640471408147, 0,
1.88551191967626174640471408147, 2.63241719874562319162363862139, 3.90673928877008595355081213073, 4.26394353215895716922063400668, 5.05664007916021817510124073443, 5.95827570516799109585872795691, 6.22115223084552657718728570847, 7.19964575265856971764462979398, 7.77392730988486122299209538619
