| L(s) = 1 | − 2-s + 4-s − 8-s − 3·11-s − 4·13-s + 16-s + 6·17-s − 2·19-s + 3·22-s + 3·23-s + 4·26-s + 9·29-s + 4·31-s − 32-s − 6·34-s + 10·37-s + 2·38-s − 5·43-s − 3·44-s − 3·46-s − 4·52-s − 3·53-s − 9·58-s + 6·59-s − 8·61-s − 4·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.904·11-s − 1.10·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.639·22-s + 0.625·23-s + 0.784·26-s + 1.67·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.324·38-s − 0.762·43-s − 0.452·44-s − 0.442·46-s − 0.554·52-s − 0.412·53-s − 1.18·58-s + 0.781·59-s − 1.02·61-s − 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008739664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008739664\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−12.96890021708583, −12.53696036872313, −12.02413156424373, −11.75911150032608, −11.12614764418637, −10.49101161971426, −10.22158526665557, −9.897682171909382, −9.319228868981211, −8.850412273706748, −8.173169193778302, −7.758265899429178, −7.663407122113743, −6.803013644120757, −6.461666296446496, −5.843517646470003, −5.249589050866198, −4.771402584919170, −4.351543334799464, −3.274168043358070, −3.034011896282092, −2.413718511504140, −1.862875809429970, −0.8867416541956017, −0.5759252482518035,
0.5759252482518035, 0.8867416541956017, 1.862875809429970, 2.413718511504140, 3.034011896282092, 3.274168043358070, 4.351543334799464, 4.771402584919170, 5.249589050866198, 5.843517646470003, 6.461666296446496, 6.803013644120757, 7.663407122113743, 7.758265899429178, 8.173169193778302, 8.850412273706748, 9.319228868981211, 9.897682171909382, 10.22158526665557, 10.49101161971426, 11.12614764418637, 11.75911150032608, 12.02413156424373, 12.53696036872313, 12.96890021708583
