| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 11-s − 13-s − 15-s + 2·17-s − 4·19-s + 4·21-s + 25-s + 27-s + 2·29-s − 8·31-s − 33-s − 4·35-s − 6·37-s − 39-s − 6·41-s + 4·43-s − 45-s + 9·49-s + 2·51-s − 2·53-s + 55-s − 4·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.174·33-s − 0.676·35-s − 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s + 0.280·51-s − 0.274·53-s + 0.134·55-s − 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.04196196445151, −14.69885356157762, −14.30595133538661, −13.86183569043649, −13.04776696673878, −12.69998716810455, −12.00196457901148, −11.57265258907446, −10.96842663646225, −10.51783399829271, −9.987193047609642, −9.177004025759796, −8.641479040200660, −8.180682150357722, −7.833467339816307, −7.116130277887381, −6.743472218061603, −5.560377285369608, −5.289999386921154, −4.509547858308840, −4.033073908750068, −3.346766291698491, −2.478468436890262, −1.886259533022104, −1.196594140242797, 0,
1.196594140242797, 1.886259533022104, 2.478468436890262, 3.346766291698491, 4.033073908750068, 4.509547858308840, 5.289999386921154, 5.560377285369608, 6.743472218061603, 7.116130277887381, 7.833467339816307, 8.180682150357722, 8.641479040200660, 9.177004025759796, 9.987193047609642, 10.51783399829271, 10.96842663646225, 11.57265258907446, 12.00196457901148, 12.69998716810455, 13.04776696673878, 13.86183569043649, 14.30595133538661, 14.69885356157762, 15.04196196445151
