| L(s) = 1 | + 1.56·5-s + 1.56·7-s + 5.56·11-s + 3.12·13-s − 6.68·17-s − 19-s + 9.12·23-s − 2.56·25-s + 8.24·29-s + 2·31-s + 2.43·35-s − 8·37-s + 5.12·41-s + 1.56·43-s − 6.68·47-s − 4.56·49-s − 4.24·53-s + 8.68·55-s + 12·59-s + 6.68·61-s + 4.87·65-s − 6.24·67-s − 16.9·73-s + 8.68·77-s − 11.3·79-s + 4·83-s − 10.4·85-s + ⋯ |
| L(s) = 1 | + 0.698·5-s + 0.590·7-s + 1.67·11-s + 0.866·13-s − 1.62·17-s − 0.229·19-s + 1.90·23-s − 0.512·25-s + 1.53·29-s + 0.359·31-s + 0.412·35-s − 1.31·37-s + 0.800·41-s + 0.238·43-s − 0.975·47-s − 0.651·49-s − 0.583·53-s + 1.17·55-s + 1.56·59-s + 0.855·61-s + 0.604·65-s − 0.763·67-s − 1.98·73-s + 0.989·77-s − 1.27·79-s + 0.439·83-s − 1.13·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.625189923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.625189923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 6.68T + 17T^{2} \) |
| 23 | \( 1 - 9.12T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 - 1.56T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 6.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−8.780530311481454533284308407795, −8.407125021088330010015534185003, −6.99774550723504708137862749777, −6.61853467458551767799179141010, −5.87766485766901116335377328835, −4.79891104823927687026714235735, −4.20090984279139781673494724554, −3.11114301211478237098068286625, −1.92426994900820057172676529012, −1.12267662912189207278137653634,
1.12267662912189207278137653634, 1.92426994900820057172676529012, 3.11114301211478237098068286625, 4.20090984279139781673494724554, 4.79891104823927687026714235735, 5.87766485766901116335377328835, 6.61853467458551767799179141010, 6.99774550723504708137862749777, 8.407125021088330010015534185003, 8.780530311481454533284308407795
