| L(s) = 1 | − 2.76·3-s + 4.24·5-s + 2.37·7-s + 4.63·9-s + 0.178·11-s − 11.7·15-s + 1.03·17-s + 5.25·19-s − 6.56·21-s + 0.130·23-s + 13.0·25-s − 4.52·27-s + 8.33·29-s + 2.94·31-s − 0.494·33-s + 10.0·35-s − 6.03·37-s − 0.264·41-s − 0.564·43-s + 19.7·45-s − 10.9·47-s − 1.36·49-s − 2.87·51-s + 2.41·53-s + 0.760·55-s − 14.5·57-s + 7.35·59-s + ⋯ |
| L(s) = 1 | − 1.59·3-s + 1.90·5-s + 0.897·7-s + 1.54·9-s + 0.0539·11-s − 3.03·15-s + 0.252·17-s + 1.20·19-s − 1.43·21-s + 0.0271·23-s + 2.61·25-s − 0.871·27-s + 1.54·29-s + 0.529·31-s − 0.0860·33-s + 1.70·35-s − 0.992·37-s − 0.0413·41-s − 0.0861·43-s + 2.93·45-s − 1.59·47-s − 0.194·49-s − 0.402·51-s + 0.332·53-s + 0.102·55-s − 1.92·57-s + 0.957·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.898684584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.898684584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 0.178T + 11T^{2} \) |
| 17 | \( 1 - 1.03T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 23 | \( 1 - 0.130T + 23T^{2} \) |
| 29 | \( 1 - 8.33T + 29T^{2} \) |
| 31 | \( 1 - 2.94T + 31T^{2} \) |
| 37 | \( 1 + 6.03T + 37T^{2} \) |
| 41 | \( 1 + 0.264T + 41T^{2} \) |
| 43 | \( 1 + 0.564T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2.41T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 + 5.06T + 61T^{2} \) |
| 67 | \( 1 - 1.58T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 3.63T + 97T^{2} \) |
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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
−9.023040954832512457164632680347, −8.053650654615968769488361210523, −6.90530367604280605867303404472, −6.41859426098273627616223350967, −5.63463201912198505609226676440, −5.15566355911287136077186686048, −4.63256367987969710671163544977, −2.96103997637910477344852033986, −1.71393808287302505058481695129, −1.04303182035722041540687006314,
1.04303182035722041540687006314, 1.71393808287302505058481695129, 2.96103997637910477344852033986, 4.63256367987969710671163544977, 5.15566355911287136077186686048, 5.63463201912198505609226676440, 6.41859426098273627616223350967, 6.90530367604280605867303404472, 8.053650654615968769488361210523, 9.023040954832512457164632680347
