| L(s) = 1 | − 2·3-s − 4·5-s + 7-s + 9-s + 4·11-s + 4·13-s + 8·15-s − 17-s + 6·19-s − 2·21-s + 11·25-s + 4·27-s − 6·29-s + 4·31-s − 8·33-s − 4·35-s + 10·37-s − 8·39-s + 6·41-s − 4·45-s + 4·47-s + 49-s + 2·51-s − 14·53-s − 16·55-s − 12·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 2.06·15-s − 0.242·17-s + 1.37·19-s − 0.436·21-s + 11/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.39·33-s − 0.676·35-s + 1.64·37-s − 1.28·39-s + 0.937·41-s − 0.596·45-s + 0.583·47-s + 1/7·49-s + 0.280·51-s − 1.92·53-s − 2.15·55-s − 1.58·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076175175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076175175\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.75639594115666082040298315776, −7.22100164091836105767301870489, −6.43356861069958630904836572455, −5.86826512030693283608532823270, −4.99983444015741342855485527468, −4.25161068988748760551155391351, −3.80594747780307910448115771540, −2.94702452529511206960076950609, −1.26776859014488952425135039834, −0.65287185429590980374299291422,
0.65287185429590980374299291422, 1.26776859014488952425135039834, 2.94702452529511206960076950609, 3.80594747780307910448115771540, 4.25161068988748760551155391351, 4.99983444015741342855485527468, 5.86826512030693283608532823270, 6.43356861069958630904836572455, 7.22100164091836105767301870489, 7.75639594115666082040298315776
