L(s) = 1 | − 3·3-s − 4·5-s − 4·7-s + 6·9-s − 2·11-s − 2·13-s + 12·15-s − 8·17-s − 6·19-s + 12·21-s − 5·23-s + 11·25-s − 9·27-s − 6·29-s + 3·31-s + 6·33-s + 16·35-s − 10·37-s + 6·39-s − 10·43-s − 24·45-s − 9·47-s + 9·49-s + 24·51-s − 11·53-s + 8·55-s + 18·57-s + ⋯ |
L(s) = 1 | − 1.73·3-s − 1.78·5-s − 1.51·7-s + 2·9-s − 0.603·11-s − 0.554·13-s + 3.09·15-s − 1.94·17-s − 1.37·19-s + 2.61·21-s − 1.04·23-s + 11/5·25-s − 1.73·27-s − 1.11·29-s + 0.538·31-s + 1.04·33-s + 2.70·35-s − 1.64·37-s + 0.960·39-s − 1.52·43-s − 3.57·45-s − 1.31·47-s + 9/7·49-s + 3.36·51-s − 1.51·53-s + 1.07·55-s + 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7537 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.97949988954495, −15.58575196513895, −15.25688452018896, −14.59611224158543, −13.36199158063591, −13.08233238789569, −12.63951074332266, −12.04606545704859, −11.81575562310738, −11.03374676236667, −10.84342070367097, −10.20855995233503, −9.680576767901322, −8.810767933335960, −8.275376657188046, −7.513866880189252, −6.937667019644035, −6.488679487289752, −6.226156822132417, −5.160997308158546, −4.725155990781252, −4.101134082349192, −3.632864472763103, −2.738409063015659, −1.713101019838388, 0, 0, 0,
1.713101019838388, 2.738409063015659, 3.632864472763103, 4.101134082349192, 4.725155990781252, 5.160997308158546, 6.226156822132417, 6.488679487289752, 6.937667019644035, 7.513866880189252, 8.275376657188046, 8.810767933335960, 9.680576767901322, 10.20855995233503, 10.84342070367097, 11.03374676236667, 11.81575562310738, 12.04606545704859, 12.63951074332266, 13.08233238789569, 13.36199158063591, 14.59611224158543, 15.25688452018896, 15.58575196513895, 15.97949988954495