| L(s) = 1 | − 0.330·3-s − 4.31·5-s − 2.89·9-s − 11-s + 5.59·13-s + 1.42·15-s + 1.52·17-s + 2.16·19-s − 7.08·23-s + 13.5·25-s + 1.94·27-s − 0.781·29-s + 5.98·31-s + 0.330·33-s − 2.86·37-s − 1.84·39-s − 6.03·41-s + 10.0·43-s + 12.4·45-s + 9.96·47-s − 0.502·51-s − 6.68·53-s + 4.31·55-s − 0.716·57-s − 3.44·59-s + 3.20·61-s − 24.0·65-s + ⋯ |
| L(s) = 1 | − 0.190·3-s − 1.92·5-s − 0.963·9-s − 0.301·11-s + 1.55·13-s + 0.367·15-s + 0.369·17-s + 0.497·19-s − 1.47·23-s + 2.71·25-s + 0.374·27-s − 0.145·29-s + 1.07·31-s + 0.0575·33-s − 0.470·37-s − 0.295·39-s − 0.942·41-s + 1.53·43-s + 1.85·45-s + 1.45·47-s − 0.0704·51-s − 0.918·53-s + 0.581·55-s − 0.0948·57-s − 0.448·59-s + 0.410·61-s − 2.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 0.330T + 3T^{2} \) |
| 5 | \( 1 + 4.31T + 5T^{2} \) |
| 13 | \( 1 - 5.59T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 + 0.781T + 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 + 6.03T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 3.44T + 59T^{2} \) |
| 61 | \( 1 - 3.20T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 + 6.26T + 71T^{2} \) |
| 73 | \( 1 - 4.35T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 1.26T + 89T^{2} \) |
| 97 | \( 1 - 0.862T + 97T^{2} \) |
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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
−7.966493928222038390295758398105, −7.58528251166981438950965826304, −6.51248748935865504884643011426, −5.87429278098197148254879044989, −4.95794419790143299347034640438, −4.02414769753470807237962091812, −3.55172585467427437565605677596, −2.72173565206530322499798392925, −1.07164935238326991796968450242, 0,
1.07164935238326991796968450242, 2.72173565206530322499798392925, 3.55172585467427437565605677596, 4.02414769753470807237962091812, 4.95794419790143299347034640438, 5.87429278098197148254879044989, 6.51248748935865504884643011426, 7.58528251166981438950965826304, 7.966493928222038390295758398105
