| L(s) = 1 | + 0.561·5-s + 11-s + 2·13-s − 7.12·17-s + 1.12·19-s − 7.68·23-s − 4.68·25-s − 7.12·29-s + 5.43·31-s − 5.68·37-s + 8.24·41-s + 1.12·43-s − 4·47-s − 7·49-s − 8.24·53-s + 0.561·55-s − 0.315·59-s + 9.36·61-s + 1.12·65-s − 7.68·67-s + 15.6·71-s − 6·73-s + 13.1·79-s − 11.3·83-s − 4·85-s − 0.561·89-s + 0.630·95-s + ⋯ |
| L(s) = 1 | + 0.251·5-s + 0.301·11-s + 0.554·13-s − 1.72·17-s + 0.257·19-s − 1.60·23-s − 0.936·25-s − 1.32·29-s + 0.976·31-s − 0.934·37-s + 1.28·41-s + 0.171·43-s − 0.583·47-s − 49-s − 1.13·53-s + 0.0757·55-s − 0.0410·59-s + 1.19·61-s + 0.139·65-s − 0.938·67-s + 1.86·71-s − 0.702·73-s + 1.47·79-s − 1.24·83-s − 0.433·85-s − 0.0595·89-s + 0.0647·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 7.68T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 + 0.315T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 - 15.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 0.561T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−8.270014063154563479463727586111, −7.67157290009868726354504921527, −6.59283119029711512261613926079, −6.19121182794177442485949515601, −5.28596208732442966384738442766, −4.27533101352841133875950963983, −3.69127476549459434347909838450, −2.41524788257273881335545028224, −1.62732435612824663355192277690, 0,
1.62732435612824663355192277690, 2.41524788257273881335545028224, 3.69127476549459434347909838450, 4.27533101352841133875950963983, 5.28596208732442966384738442766, 6.19121182794177442485949515601, 6.59283119029711512261613926079, 7.67157290009868726354504921527, 8.270014063154563479463727586111
