| L(s) = 1 | + 2.44·2-s − 2.96·3-s + 3.99·4-s − 5-s − 7.26·6-s + 4.87·8-s + 5.81·9-s − 2.44·10-s − 0.392·11-s − 11.8·12-s + 2.11·13-s + 2.96·15-s + 3.94·16-s − 3.76·17-s + 14.2·18-s − 0.490·19-s − 3.99·20-s − 0.960·22-s + 1.45·23-s − 14.4·24-s + 25-s + 5.17·26-s − 8.35·27-s − 1.63·29-s + 7.26·30-s + 3.12·31-s − 0.0823·32-s + ⋯ |
| L(s) = 1 | + 1.73·2-s − 1.71·3-s + 1.99·4-s − 0.447·5-s − 2.96·6-s + 1.72·8-s + 1.93·9-s − 0.774·10-s − 0.118·11-s − 3.42·12-s + 0.585·13-s + 0.766·15-s + 0.987·16-s − 0.912·17-s + 3.35·18-s − 0.112·19-s − 0.892·20-s − 0.204·22-s + 0.302·23-s − 2.95·24-s + 0.200·25-s + 1.01·26-s − 1.60·27-s − 0.303·29-s + 1.32·30-s + 0.560·31-s − 0.0145·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 11 | \( 1 + 0.392T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 0.490T + 19T^{2} \) |
| 23 | \( 1 - 1.45T + 23T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 0.921T + 43T^{2} \) |
| 47 | \( 1 + 3.30T + 47T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + 5.24T + 59T^{2} \) |
| 61 | \( 1 - 9.18T + 61T^{2} \) |
| 67 | \( 1 + 6.32T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 6.61T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 - 5.78T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 - 4.07T + 97T^{2} \) |
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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
−6.84768315708917921191152518924, −6.46343435634346314194229758163, −5.94624254389336037867338751252, −5.12590208251859276651932147614, −4.82637733999504246030462408486, −4.08225647984455714497593382498, −3.47024723929464219477186573403, −2.37644738203418625524331354293, −1.29943938781582045688383529792, 0,
1.29943938781582045688383529792, 2.37644738203418625524331354293, 3.47024723929464219477186573403, 4.08225647984455714497593382498, 4.82637733999504246030462408486, 5.12590208251859276651932147614, 5.94624254389336037867338751252, 6.46343435634346314194229758163, 6.84768315708917921191152518924
