| L(s) = 1 | + 0.0697·3-s + 2.59·5-s − 2.99·9-s + 4.00·11-s − 13-s + 0.180·15-s + 1.86·17-s − 6.43·19-s − 1.24·23-s + 1.72·25-s − 0.418·27-s − 8.09·29-s − 8.57·31-s + 0.279·33-s − 6.60·37-s − 0.0697·39-s − 5.25·41-s − 6.20·43-s − 7.76·45-s + 9.73·47-s + 0.129·51-s − 10.3·53-s + 10.3·55-s − 0.448·57-s − 13.3·59-s + 11.0·61-s − 2.59·65-s + ⋯ |
| L(s) = 1 | + 0.0402·3-s + 1.15·5-s − 0.998·9-s + 1.20·11-s − 0.277·13-s + 0.0466·15-s + 0.451·17-s − 1.47·19-s − 0.260·23-s + 0.344·25-s − 0.0804·27-s − 1.50·29-s − 1.53·31-s + 0.0486·33-s − 1.08·37-s − 0.0111·39-s − 0.820·41-s − 0.946·43-s − 1.15·45-s + 1.41·47-s + 0.0181·51-s − 1.42·53-s + 1.39·55-s − 0.0594·57-s − 1.73·59-s + 1.41·61-s − 0.321·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0697T + 3T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 - 4.00T + 11T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 8.09T + 29T^{2} \) |
| 31 | \( 1 + 8.57T + 31T^{2} \) |
| 37 | \( 1 + 6.60T + 37T^{2} \) |
| 41 | \( 1 + 5.25T + 41T^{2} \) |
| 43 | \( 1 + 6.20T + 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.28T + 67T^{2} \) |
| 71 | \( 1 - 0.980T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 0.218T + 83T^{2} \) |
| 89 | \( 1 + 9.29T + 89T^{2} \) |
| 97 | \( 1 + 8.61T + 97T^{2} \) |
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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
−7.966499072235090068763875465369, −6.96500590732145989035794451486, −6.36649402294091110278794965747, −5.70448179041236254885445086372, −5.18260705754399564958712865049, −4.01687139740469627800684961271, −3.33018767751737112052641140356, −2.14476380727482255160946348217, −1.66731265261821828615797364172, 0,
1.66731265261821828615797364172, 2.14476380727482255160946348217, 3.33018767751737112052641140356, 4.01687139740469627800684961271, 5.18260705754399564958712865049, 5.70448179041236254885445086372, 6.36649402294091110278794965747, 6.96500590732145989035794451486, 7.966499072235090068763875465369
