| L(s) = 1 | − 3-s + 3·5-s + 9-s + 3·11-s − 3·15-s − 17-s − 19-s + 4·25-s − 27-s + 8·29-s − 2·31-s − 3·33-s + 4·37-s + 12·41-s − 43-s + 3·45-s + 9·47-s + 51-s − 6·53-s + 9·55-s + 57-s + 6·59-s − 61-s − 10·67-s + 10·71-s + 11·73-s − 4·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.904·11-s − 0.774·15-s − 0.242·17-s − 0.229·19-s + 4/5·25-s − 0.192·27-s + 1.48·29-s − 0.359·31-s − 0.522·33-s + 0.657·37-s + 1.87·41-s − 0.152·43-s + 0.447·45-s + 1.31·47-s + 0.140·51-s − 0.824·53-s + 1.21·55-s + 0.132·57-s + 0.781·59-s − 0.128·61-s − 1.22·67-s + 1.18·71-s + 1.28·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.944532721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.944532721\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.00742248339072, −12.79416031249157, −12.29184105518398, −11.70463602863687, −11.30583054062324, −10.76037169816844, −10.29954428986946, −9.916768750900867, −9.376130883012749, −8.994520584082577, −8.599953583329459, −7.730699225304449, −7.373900316329071, −6.554335853528354, −6.339840327328094, −5.975089119614430, −5.395205215403735, −4.816967149279466, −4.334744343943496, −3.769184335313694, −2.966275346825013, −2.325494423253852, −1.889899599937571, −1.087737677004449, −0.6718289625227662,
0.6718289625227662, 1.087737677004449, 1.889899599937571, 2.325494423253852, 2.966275346825013, 3.769184335313694, 4.334744343943496, 4.816967149279466, 5.395205215403735, 5.975089119614430, 6.339840327328094, 6.554335853528354, 7.373900316329071, 7.730699225304449, 8.599953583329459, 8.994520584082577, 9.376130883012749, 9.916768750900867, 10.29954428986946, 10.76037169816844, 11.30583054062324, 11.70463602863687, 12.29184105518398, 12.79416031249157, 13.00742248339072
