| L(s) = 1 | − 3·3-s + 6·9-s − 13-s + 2·17-s − 2·19-s − 6·23-s − 5·25-s − 9·27-s + 29-s − 4·31-s − 2·37-s + 3·39-s − 2·41-s + 8·43-s + 6·47-s − 6·51-s − 8·53-s + 6·57-s + 59-s − 5·61-s − 9·67-s + 18·69-s − 4·71-s − 10·73-s + 15·75-s + 9·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 0.277·13-s + 0.485·17-s − 0.458·19-s − 1.25·23-s − 25-s − 1.73·27-s + 0.185·29-s − 0.718·31-s − 0.328·37-s + 0.480·39-s − 0.312·41-s + 1.21·43-s + 0.875·47-s − 0.840·51-s − 1.09·53-s + 0.794·57-s + 0.130·59-s − 0.640·61-s − 1.09·67-s + 2.16·69-s − 0.474·71-s − 1.17·73-s + 1.73·75-s + 1.01·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4190025848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4190025848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−12.98862246670275, −12.37282257974678, −12.06297177026528, −11.91266835522876, −11.20101884932890, −10.75191161167770, −10.48785000291610, −9.919414149440611, −9.521698832265548, −8.956531999048443, −8.218289400425728, −7.643387229808396, −7.341182445411393, −6.703880138055256, −6.148048744744709, −5.794598036029986, −5.518332842448106, −4.694611298922524, −4.440713468475350, −3.818245736789651, −3.197381851545405, −2.209201809808318, −1.756878189488406, −0.9960514325221239, −0.2391176523518854,
0.2391176523518854, 0.9960514325221239, 1.756878189488406, 2.209201809808318, 3.197381851545405, 3.818245736789651, 4.440713468475350, 4.694611298922524, 5.518332842448106, 5.794598036029986, 6.148048744744709, 6.703880138055256, 7.341182445411393, 7.643387229808396, 8.218289400425728, 8.956531999048443, 9.521698832265548, 9.919414149440611, 10.48785000291610, 10.75191161167770, 11.20101884932890, 11.91266835522876, 12.06297177026528, 12.37282257974678, 12.98862246670275
