| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s − 2·11-s − 12-s + 2·13-s + 16-s − 2·18-s − 2·19-s − 2·22-s + 23-s − 24-s + 2·26-s + 5·27-s + 5·29-s − 2·31-s + 32-s + 2·33-s − 2·36-s − 2·37-s − 2·38-s − 2·39-s + 9·41-s + 43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.603·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.471·18-s − 0.458·19-s − 0.426·22-s + 0.208·23-s − 0.204·24-s + 0.392·26-s + 0.962·27-s + 0.928·29-s − 0.359·31-s + 0.176·32-s + 0.348·33-s − 1/3·36-s − 0.328·37-s − 0.324·38-s − 0.320·39-s + 1.40·41-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.967954122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967954122\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−12.96413525913615, −12.67272277669333, −12.23695124371615, −11.62237348244318, −11.31775791293517, −10.81919500167319, −10.49985033507990, −9.998473373401258, −9.242724405604115, −8.811844597692038, −8.261463625640348, −7.793311787205162, −7.268644328368681, −6.586302146762412, −6.149823800508570, −5.924087797458503, −5.123106575089774, −4.916940612150632, −4.330402914969180, −3.521730657417203, −3.249554897948222, −2.424188032230138, −2.085957016618707, −1.064711293738118, −0.4959084557458784,
0.4959084557458784, 1.064711293738118, 2.085957016618707, 2.424188032230138, 3.249554897948222, 3.521730657417203, 4.330402914969180, 4.916940612150632, 5.123106575089774, 5.924087797458503, 6.149823800508570, 6.586302146762412, 7.268644328368681, 7.793311787205162, 8.261463625640348, 8.811844597692038, 9.242724405604115, 9.998473373401258, 10.49985033507990, 10.81919500167319, 11.31775791293517, 11.62237348244318, 12.23695124371615, 12.67272277669333, 12.96413525913615
