| L(s) = 1 | + 5-s − 4·7-s − 6·11-s + 4·13-s − 3·17-s + 7·19-s − 9·23-s + 25-s − 7·31-s − 4·35-s − 2·37-s + 6·41-s − 2·43-s + 9·49-s − 9·53-s − 6·55-s + 12·59-s + 7·61-s + 4·65-s − 2·67-s + 6·71-s + 2·73-s + 24·77-s − 79-s − 9·83-s − 3·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 1.80·11-s + 1.10·13-s − 0.727·17-s + 1.60·19-s − 1.87·23-s + 1/5·25-s − 1.25·31-s − 0.676·35-s − 0.328·37-s + 0.937·41-s − 0.304·43-s + 9/7·49-s − 1.23·53-s − 0.809·55-s + 1.56·59-s + 0.896·61-s + 0.496·65-s − 0.244·67-s + 0.712·71-s + 0.234·73-s + 2.73·77-s − 0.112·79-s − 0.987·83-s − 0.325·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.062520269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.062520269\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.75445560307162518433276120482, −7.05385618921565081755289977771, −6.31428142780564962629578498673, −5.68977345778808165179879459440, −5.28799647162051247248572387731, −4.08851044485322199201440937509, −3.38504060558484704716719946533, −2.72890820492153798397328653865, −1.88198462804690228011401538227, −0.47605480509136641046831994616,
0.47605480509136641046831994616, 1.88198462804690228011401538227, 2.72890820492153798397328653865, 3.38504060558484704716719946533, 4.08851044485322199201440937509, 5.28799647162051247248572387731, 5.68977345778808165179879459440, 6.31428142780564962629578498673, 7.05385618921565081755289977771, 7.75445560307162518433276120482
