| L(s) = 1 | + 2·2-s + 2·4-s − 5-s + 7-s − 2·10-s − 11-s + 4·13-s + 2·14-s − 4·16-s − 8·17-s − 19-s − 2·20-s − 2·22-s + 6·23-s − 4·25-s + 8·26-s + 2·28-s + 7·31-s − 8·32-s − 16·34-s − 35-s + 8·37-s − 2·38-s + 3·41-s + 4·43-s − 2·44-s + 12·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s + 0.377·7-s − 0.632·10-s − 0.301·11-s + 1.10·13-s + 0.534·14-s − 16-s − 1.94·17-s − 0.229·19-s − 0.447·20-s − 0.426·22-s + 1.25·23-s − 4/5·25-s + 1.56·26-s + 0.377·28-s + 1.25·31-s − 1.41·32-s − 2.74·34-s − 0.169·35-s + 1.31·37-s − 0.324·38-s + 0.468·41-s + 0.609·43-s − 0.301·44-s + 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13167 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13167 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−16.09475108734683, −15.72875640513512, −15.30985429255333, −14.93839428324990, −14.06956972894331, −13.72976879708229, −13.07579437542914, −12.88378018598806, −12.04760470384266, −11.41344373080417, −11.07133877273049, −10.65687506698783, −9.443611344447920, −9.002216350120502, −8.284748017922372, −7.694106399018996, −6.776600331054879, −6.313034572858180, −5.776376786268113, −4.811758150251550, −4.452843847292708, −3.932498702536318, −3.003074772109791, −2.476126352277646, −1.378992109896953, 0,
1.378992109896953, 2.476126352277646, 3.003074772109791, 3.932498702536318, 4.452843847292708, 4.811758150251550, 5.776376786268113, 6.313034572858180, 6.776600331054879, 7.694106399018996, 8.284748017922372, 9.002216350120502, 9.443611344447920, 10.65687506698783, 11.07133877273049, 11.41344373080417, 12.04760470384266, 12.88378018598806, 13.07579437542914, 13.72976879708229, 14.06956972894331, 14.93839428324990, 15.30985429255333, 15.72875640513512, 16.09475108734683
