| L(s) = 1 | − 2·3-s − 5-s + 2·7-s + 9-s + 4·11-s + 2·13-s + 2·15-s + 6·17-s − 4·21-s − 2·23-s + 25-s + 4·27-s + 29-s + 4·31-s − 8·33-s − 2·35-s − 2·37-s − 4·39-s + 6·41-s + 6·43-s − 45-s + 6·47-s − 3·49-s − 12·51-s − 14·53-s − 4·55-s − 6·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.872·21-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 0.185·29-s + 0.718·31-s − 1.39·33-s − 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.937·41-s + 0.914·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 1.68·51-s − 1.92·53-s − 0.539·55-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470536602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470536602\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−8.151188877472253761812477127186, −7.65356456870670373503263832476, −6.67297771047602745577747884913, −6.09716285025677175970416577734, −5.45876542559161291193321247475, −4.63535673962995268644858743671, −3.97298582848090646968916654007, −3.02569089968049578451374474875, −1.53134035306265294644800935703, −0.790471383193495901600322060210,
0.790471383193495901600322060210, 1.53134035306265294644800935703, 3.02569089968049578451374474875, 3.97298582848090646968916654007, 4.63535673962995268644858743671, 5.45876542559161291193321247475, 6.09716285025677175970416577734, 6.67297771047602745577747884913, 7.65356456870670373503263832476, 8.151188877472253761812477127186
