| L(s) = 1 | − 0.0950·3-s − 1.17·5-s + 3.14·7-s − 2.99·9-s + 1.67·11-s − 6.63·13-s + 0.111·15-s + 5.16·17-s + 4.72·19-s − 0.299·21-s − 8.82·23-s − 3.63·25-s + 0.569·27-s + 1.80·29-s − 1.65·31-s − 0.159·33-s − 3.68·35-s + 1.99·37-s + 0.630·39-s − 41-s − 1.46·43-s + 3.50·45-s − 8.53·47-s + 2.89·49-s − 0.490·51-s + 9.35·53-s − 1.96·55-s + ⋯ |
| L(s) = 1 | − 0.0549·3-s − 0.523·5-s + 1.18·7-s − 0.996·9-s + 0.505·11-s − 1.83·13-s + 0.0287·15-s + 1.25·17-s + 1.08·19-s − 0.0652·21-s − 1.83·23-s − 0.726·25-s + 0.109·27-s + 0.336·29-s − 0.298·31-s − 0.0277·33-s − 0.622·35-s + 0.327·37-s + 0.100·39-s − 0.156·41-s − 0.224·43-s + 0.521·45-s − 1.24·47-s + 0.413·49-s − 0.0687·51-s + 1.28·53-s − 0.264·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0950T + 3T^{2} \) |
| 5 | \( 1 + 1.17T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 6.63T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 - 1.80T + 29T^{2} \) |
| 31 | \( 1 + 1.65T + 31T^{2} \) |
| 37 | \( 1 - 1.99T + 37T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 - 9.35T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 9.67T + 67T^{2} \) |
| 71 | \( 1 + 0.776T + 71T^{2} \) |
| 73 | \( 1 - 8.33T + 73T^{2} \) |
| 79 | \( 1 - 0.915T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−8.055362869403645867942928116500, −7.983370533100929973844825867688, −7.22088666863245441130452274409, −6.00880883878907367762272700938, −5.30502774516938323048043522920, −4.62959974407541175858918861065, −3.64246933599559843307527313298, −2.64521256393266791491561857519, −1.55545745047857189007662002067, 0,
1.55545745047857189007662002067, 2.64521256393266791491561857519, 3.64246933599559843307527313298, 4.62959974407541175858918861065, 5.30502774516938323048043522920, 6.00880883878907367762272700938, 7.22088666863245441130452274409, 7.983370533100929973844825867688, 8.055362869403645867942928116500
