| L(s) = 1 | − 3.98·5-s + 1.07·7-s − 2.72·11-s + 1.35·13-s − 5.62·17-s + 7.96·19-s + 0.725·23-s + 10.8·25-s − 0.488·29-s + 4.05·31-s − 4.28·35-s − 6.63·37-s − 1.03·41-s + 2.77·43-s + 2.73·47-s − 5.84·49-s + 6.80·53-s + 10.8·55-s + 8.67·59-s − 7.54·61-s − 5.41·65-s − 0.660·67-s + 9.99·71-s − 0.870·73-s − 2.92·77-s − 2.47·79-s − 4.15·83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.406·7-s − 0.820·11-s + 0.376·13-s − 1.36·17-s + 1.82·19-s + 0.151·23-s + 2.17·25-s − 0.0906·29-s + 0.728·31-s − 0.725·35-s − 1.09·37-s − 0.161·41-s + 0.422·43-s + 0.398·47-s − 0.834·49-s + 0.934·53-s + 1.46·55-s + 1.12·59-s − 0.966·61-s − 0.672·65-s − 0.0806·67-s + 1.18·71-s − 0.101·73-s − 0.333·77-s − 0.278·79-s − 0.455·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.98T + 5T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 - 0.725T + 23T^{2} \) |
| 29 | \( 1 + 0.488T + 29T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 - 2.73T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 - 8.67T + 59T^{2} \) |
| 61 | \( 1 + 7.54T + 61T^{2} \) |
| 67 | \( 1 + 0.660T + 67T^{2} \) |
| 71 | \( 1 - 9.99T + 71T^{2} \) |
| 73 | \( 1 + 0.870T + 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 + 4.15T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.64406453865404792478077577532, −7.28105617065321139820149904350, −6.49573446953267007231424053180, −5.32223707294192409693722990744, −4.81232879284452210464542907323, −4.01079911057924801826473426966, −3.34655028391822015426926905660, −2.50227902652188366619890343642, −1.09388233856811213329273638772, 0,
1.09388233856811213329273638772, 2.50227902652188366619890343642, 3.34655028391822015426926905660, 4.01079911057924801826473426966, 4.81232879284452210464542907323, 5.32223707294192409693722990744, 6.49573446953267007231424053180, 7.28105617065321139820149904350, 7.64406453865404792478077577532
