| L(s) = 1 | + 3-s − 5-s + 9-s + 6·11-s − 2·13-s − 15-s − 4·17-s − 2·23-s + 25-s + 27-s − 8·29-s − 31-s + 6·33-s − 6·37-s − 2·39-s − 2·41-s − 4·43-s − 45-s − 4·47-s − 7·49-s − 4·51-s − 6·53-s − 6·55-s + 4·61-s + 2·65-s + 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.258·15-s − 0.970·17-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 0.179·31-s + 1.04·33-s − 0.986·37-s − 0.320·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.583·47-s − 49-s − 0.560·51-s − 0.824·53-s − 0.809·55-s + 0.512·61-s + 0.248·65-s + 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.57589402440684058797457513997, −6.77218791705461446935157294475, −6.49199074004246415432638423070, −5.35099709259820822318154145004, −4.52452855995751934068401276641, −3.84305291073178410890287425764, −3.34012969458339636946505371764, −2.15297461090480767306938359552, −1.46627385451330423984937899737, 0,
1.46627385451330423984937899737, 2.15297461090480767306938359552, 3.34012969458339636946505371764, 3.84305291073178410890287425764, 4.52452855995751934068401276641, 5.35099709259820822318154145004, 6.49199074004246415432638423070, 6.77218791705461446935157294475, 7.57589402440684058797457513997
