| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s − 2·13-s − 15-s − 6·17-s + 4·19-s − 21-s + 25-s + 27-s − 6·29-s − 4·31-s + 35-s − 10·37-s − 2·39-s − 2·41-s + 4·43-s − 45-s + 49-s − 6·51-s − 53-s + 4·57-s − 8·59-s − 6·61-s − 63-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.169·35-s − 1.64·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.137·53-s + 0.529·57-s − 1.04·59-s − 0.768·61-s − 0.125·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.205173589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.205173589\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.75698899929081, −14.04413333412106, −13.69276560466509, −13.17406951821675, −12.55354072519546, −12.21904000339497, −11.55418689368954, −10.91073032151664, −10.64672495249033, −9.721990762464258, −9.405352206992613, −8.914161416465174, −8.344774064488705, −7.677223595202642, −7.226028910123366, −6.772185604770302, −6.104515054186187, −5.223731525095730, −4.876420093220850, −3.944473860900148, −3.637974819810262, −2.874508673212850, −2.212026632628264, −1.539625104390173, −0.3655606425781235,
0.3655606425781235, 1.539625104390173, 2.212026632628264, 2.874508673212850, 3.637974819810262, 3.944473860900148, 4.876420093220850, 5.223731525095730, 6.104515054186187, 6.772185604770302, 7.226028910123366, 7.677223595202642, 8.344774064488705, 8.914161416465174, 9.405352206992613, 9.721990762464258, 10.64672495249033, 10.91073032151664, 11.55418689368954, 12.21904000339497, 12.55354072519546, 13.17406951821675, 13.69276560466509, 14.04413333412106, 14.75698899929081
