| L(s) = 1 | + 2.54·3-s + 3.21·5-s + 1.46·7-s + 3.46·9-s − 1.87·11-s + 3.02·13-s + 8.18·15-s − 17-s − 3.37·19-s + 3.72·21-s − 0.300·23-s + 5.36·25-s + 1.18·27-s + 7.78·29-s + 0.320·31-s − 4.76·33-s + 4.71·35-s − 2.10·37-s + 7.69·39-s − 9.22·41-s + 11.4·43-s + 11.1·45-s + 7.32·47-s − 4.85·49-s − 2.54·51-s + 6.35·53-s − 6.02·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s + 1.43·5-s + 0.553·7-s + 1.15·9-s − 0.564·11-s + 0.839·13-s + 2.11·15-s − 0.242·17-s − 0.775·19-s + 0.813·21-s − 0.0626·23-s + 1.07·25-s + 0.227·27-s + 1.44·29-s + 0.0575·31-s − 0.828·33-s + 0.797·35-s − 0.345·37-s + 1.23·39-s − 1.44·41-s + 1.75·43-s + 1.66·45-s + 1.06·47-s − 0.693·49-s − 0.356·51-s + 0.872·53-s − 0.812·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.687148568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.687148568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 - 3.21T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 - 3.02T + 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 + 0.300T + 23T^{2} \) |
| 29 | \( 1 - 7.78T + 29T^{2} \) |
| 31 | \( 1 - 0.320T + 31T^{2} \) |
| 37 | \( 1 + 2.10T + 37T^{2} \) |
| 41 | \( 1 + 9.22T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 - 6.35T + 53T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 5.00T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 + 3.87T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−8.489615615475783621563610067673, −8.029020199775867460244758267417, −7.03324448651109590634381865876, −6.26090145157715128177274048071, −5.50219750579599383346342074469, −4.61100435910367936815272009065, −3.69683933456613331620802585020, −2.66583723953046687530489001520, −2.19139955351729488653531988795, −1.30557374823205282487648539098,
1.30557374823205282487648539098, 2.19139955351729488653531988795, 2.66583723953046687530489001520, 3.69683933456613331620802585020, 4.61100435910367936815272009065, 5.50219750579599383346342074469, 6.26090145157715128177274048071, 7.03324448651109590634381865876, 8.029020199775867460244758267417, 8.489615615475783621563610067673
