| L(s) = 1 | + 2-s + 4-s + 2·5-s − 2·7-s + 8-s + 2·10-s − 4·11-s − 2·14-s + 16-s + 2·17-s + 2·20-s − 4·22-s + 23-s − 25-s − 2·28-s − 6·29-s + 2·31-s + 32-s + 2·34-s − 4·35-s − 2·37-s + 2·40-s + 2·41-s + 4·43-s − 4·44-s + 46-s − 12·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.755·7-s + 0.353·8-s + 0.632·10-s − 1.20·11-s − 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.447·20-s − 0.852·22-s + 0.208·23-s − 1/5·25-s − 0.377·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.342·34-s − 0.676·35-s − 0.328·37-s + 0.316·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.147·46-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149454 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.962678668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.962678668\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.22435323156125, −13.03648798396610, −12.59434906398314, −12.08227258183470, −11.37992634284704, −11.05844072842022, −10.39050746251313, −10.01762836881648, −9.664633383052790, −9.141706950056806, −8.477111637496385, −7.769905122073400, −7.569582411378861, −6.760252815914556, −6.352773991383995, −5.890559639513438, −5.316715869997508, −5.077036151519321, −4.337150232879291, −3.549011904276544, −3.238058173638045, −2.515285175459499, −2.103888229702031, −1.397105341735579, −0.4234791271007763,
0.4234791271007763, 1.397105341735579, 2.103888229702031, 2.515285175459499, 3.238058173638045, 3.549011904276544, 4.337150232879291, 5.077036151519321, 5.316715869997508, 5.890559639513438, 6.352773991383995, 6.760252815914556, 7.569582411378861, 7.769905122073400, 8.477111637496385, 9.141706950056806, 9.664633383052790, 10.01762836881648, 10.39050746251313, 11.05844072842022, 11.37992634284704, 12.08227258183470, 12.59434906398314, 13.03648798396610, 13.22435323156125
