| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 4·13-s − 2·14-s + 16-s − 6·17-s − 20-s + 25-s − 4·26-s + 2·28-s − 6·29-s + 4·31-s − 32-s + 6·34-s − 2·35-s − 8·37-s + 40-s − 6·41-s − 7·43-s − 6·47-s − 3·49-s − 50-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.223·20-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.338·35-s − 1.31·37-s + 0.158·40-s − 0.937·41-s − 1.06·43-s − 0.875·47-s − 3/7·49-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 292410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 292410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6663871396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6663871396\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−12.81248233080499, −12.01409305849457, −11.52727741168141, −11.35431281442169, −11.02917421182138, −10.32815720587653, −10.11681890095028, −9.369572244836867, −8.893764442458581, −8.481948862557562, −8.241685910395961, −7.751107931528267, −7.022108809434914, −6.785306307498071, −6.263496690339058, −5.676454590816660, −4.994879550195928, −4.671735735790807, −3.958162536133919, −3.479538234442658, −2.975983467179966, −2.046673806228259, −1.771437089363973, −1.158212784732499, −0.2514836202092535,
0.2514836202092535, 1.158212784732499, 1.771437089363973, 2.046673806228259, 2.975983467179966, 3.479538234442658, 3.958162536133919, 4.671735735790807, 4.994879550195928, 5.676454590816660, 6.263496690339058, 6.785306307498071, 7.022108809434914, 7.751107931528267, 8.241685910395961, 8.481948862557562, 8.893764442458581, 9.369572244836867, 10.11681890095028, 10.32815720587653, 11.02917421182138, 11.35431281442169, 11.52727741168141, 12.01409305849457, 12.81248233080499
