| L(s) = 1 | + 5·5-s + 4·7-s − 72·11-s − 6·13-s − 38·17-s + 52·19-s − 152·23-s + 25·25-s + 78·29-s + 120·31-s + 20·35-s − 150·37-s − 362·41-s − 484·43-s − 280·47-s − 327·49-s + 670·53-s − 360·55-s − 696·59-s + 222·61-s − 30·65-s − 4·67-s − 96·71-s + 178·73-s − 288·77-s − 632·79-s + 612·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.215·7-s − 1.97·11-s − 0.128·13-s − 0.542·17-s + 0.627·19-s − 1.37·23-s + 1/5·25-s + 0.499·29-s + 0.695·31-s + 0.0965·35-s − 0.666·37-s − 1.37·41-s − 1.71·43-s − 0.868·47-s − 0.953·49-s + 1.73·53-s − 0.882·55-s − 1.53·59-s + 0.465·61-s − 0.0572·65-s − 0.00729·67-s − 0.160·71-s + 0.285·73-s − 0.426·77-s − 0.900·79-s + 0.809·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 72 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 120 T + p^{3} T^{2} \) |
| 37 | \( 1 + 150 T + p^{3} T^{2} \) |
| 41 | \( 1 + 362 T + p^{3} T^{2} \) |
| 43 | \( 1 + 484 T + p^{3} T^{2} \) |
| 47 | \( 1 + 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 670 T + p^{3} T^{2} \) |
| 59 | \( 1 + 696 T + p^{3} T^{2} \) |
| 61 | \( 1 - 222 T + p^{3} T^{2} \) |
| 67 | \( 1 + 4 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 83 | \( 1 - 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 994 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1634 T + p^{3} T^{2} \) |
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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
−10.31232220772037816188183055028, −9.957875357294756363578128538589, −8.520544665819505142148721585698, −7.86465506670239731604917762644, −6.69881459015166286111700085807, −5.51396870537993882434202467442, −4.75075603913950026754712516713, −3.10050198913070458617907559231, −1.94611594150211055675051472630, 0,
1.94611594150211055675051472630, 3.10050198913070458617907559231, 4.75075603913950026754712516713, 5.51396870537993882434202467442, 6.69881459015166286111700085807, 7.86465506670239731604917762644, 8.520544665819505142148721585698, 9.957875357294756363578128538589, 10.31232220772037816188183055028
