| L(s) = 1 | + 0.585·5-s + 3.41·7-s − 3.24·11-s − 0.585·13-s − 3.24·17-s − 0.242·19-s − 4.24·23-s − 4.65·25-s + 6.41·29-s + 7.82·31-s + 2·35-s − 2.17·37-s − 41-s + 4.17·43-s + 4.07·47-s + 4.65·49-s + 4.48·53-s − 1.89·55-s + 1.17·59-s + 8.31·61-s − 0.343·65-s + 10.4·67-s + 11.7·71-s + 3.34·73-s − 11.0·77-s + 17.3·79-s + 4.58·83-s + ⋯ |
| L(s) = 1 | + 0.261·5-s + 1.29·7-s − 0.977·11-s − 0.162·13-s − 0.786·17-s − 0.0556·19-s − 0.884·23-s − 0.931·25-s + 1.19·29-s + 1.40·31-s + 0.338·35-s − 0.357·37-s − 0.156·41-s + 0.636·43-s + 0.593·47-s + 0.665·49-s + 0.616·53-s − 0.256·55-s + 0.152·59-s + 1.06·61-s − 0.0425·65-s + 1.28·67-s + 1.39·71-s + 0.391·73-s − 1.26·77-s + 1.94·79-s + 0.503·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.172752504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.172752504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 + 0.242T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 - 7.82T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 - 4.07T + 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 - 8.31T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 3.34T + 73T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 0.928T + 97T^{2} \) |
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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
−8.272195726007632856851981542675, −7.52539533088402511435380772054, −6.67285596326200563656696843108, −5.90197797291224552049340948681, −5.10517892981073322917501819253, −4.61803507231886498796578788082, −3.77004502859711205722305660507, −2.46736427028109806713196106415, −2.06801161060073530446975087595, −0.77274291707481474329335927556,
0.77274291707481474329335927556, 2.06801161060073530446975087595, 2.46736427028109806713196106415, 3.77004502859711205722305660507, 4.61803507231886498796578788082, 5.10517892981073322917501819253, 5.90197797291224552049340948681, 6.67285596326200563656696843108, 7.52539533088402511435380772054, 8.272195726007632856851981542675
