| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·11-s + 6·13-s + 14-s + 16-s − 2·17-s + 6·19-s + 2·22-s + 23-s + 6·26-s + 28-s − 9·29-s − 2·31-s + 32-s − 2·34-s − 2·37-s + 6·38-s + 11·41-s + 4·43-s + 2·44-s + 46-s − 7·47-s − 6·49-s + 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.426·22-s + 0.208·23-s + 1.17·26-s + 0.188·28-s − 1.67·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.973·38-s + 1.71·41-s + 0.609·43-s + 0.301·44-s + 0.147·46-s − 1.02·47-s − 6/7·49-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.760270237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.760270237\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−8.388978096073027962207440543119, −7.58492416244869339589972035105, −6.91495102901194921790514761725, −6.03185539920883858332542008312, −5.56904876216617534173150872255, −4.61001389159978610074950697273, −3.79659414428051637942667244535, −3.24178636389820284326167816963, −1.96309186920039957139249194652, −1.09108696574790372712953100792,
1.09108696574790372712953100792, 1.96309186920039957139249194652, 3.24178636389820284326167816963, 3.79659414428051637942667244535, 4.61001389159978610074950697273, 5.56904876216617534173150872255, 6.03185539920883858332542008312, 6.91495102901194921790514761725, 7.58492416244869339589972035105, 8.388978096073027962207440543119
