| L(s) = 1 | − 3·5-s − 7-s − 3·9-s − 2·11-s − 4·17-s − 19-s − 7·23-s + 4·25-s − 7·29-s + 5·31-s + 3·35-s + 4·37-s + 6·41-s − 9·43-s + 9·45-s + 7·47-s + 49-s − 11·53-s + 6·55-s + 2·61-s + 3·63-s − 10·67-s − 7·73-s + 2·77-s + 79-s + 9·81-s − 11·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 9-s − 0.603·11-s − 0.970·17-s − 0.229·19-s − 1.45·23-s + 4/5·25-s − 1.29·29-s + 0.898·31-s + 0.507·35-s + 0.657·37-s + 0.937·41-s − 1.37·43-s + 1.34·45-s + 1.02·47-s + 1/7·49-s − 1.51·53-s + 0.809·55-s + 0.256·61-s + 0.377·63-s − 1.22·67-s − 0.819·73-s + 0.227·77-s + 0.112·79-s + 81-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−14.45461864936870, −13.72646540567056, −13.30147254347342, −12.82259337952359, −12.14334881971562, −11.83941378466625, −11.33511968411311, −10.92104443804169, −10.43987458937601, −9.705451817101760, −9.237588445567016, −8.523641733183708, −8.227732961992044, −7.726453423503517, −7.258549217785760, −6.549771286682929, −5.961448850577966, −5.583474292573412, −4.606073932685489, −4.337001290507574, −3.647076586748627, −3.083086413199201, −2.494980399427662, −1.780482681136548, −0.5061845841716397, 0,
0.5061845841716397, 1.780482681136548, 2.494980399427662, 3.083086413199201, 3.647076586748627, 4.337001290507574, 4.606073932685489, 5.583474292573412, 5.961448850577966, 6.549771286682929, 7.258549217785760, 7.726453423503517, 8.227732961992044, 8.523641733183708, 9.237588445567016, 9.705451817101760, 10.43987458937601, 10.92104443804169, 11.33511968411311, 11.83941378466625, 12.14334881971562, 12.82259337952359, 13.30147254347342, 13.72646540567056, 14.45461864936870
