| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·11-s + 6·13-s + 2·14-s + 16-s − 8·19-s + 2·22-s + 2·23-s − 6·26-s − 2·28-s + 6·29-s + 2·31-s − 32-s + 6·37-s + 8·38-s + 2·41-s + 4·43-s − 2·44-s − 2·46-s + 4·47-s − 3·49-s + 6·52-s − 10·53-s + 2·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 1.83·19-s + 0.426·22-s + 0.417·23-s − 1.17·26-s − 0.377·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.986·37-s + 1.29·38-s + 0.312·41-s + 0.609·43-s − 0.301·44-s − 0.294·46-s + 0.583·47-s − 3/7·49-s + 0.832·52-s − 1.37·53-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458212347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458212347\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.33471643646272, −12.84541353063839, −12.74263201520404, −12.02995728916267, −11.36634870385757, −10.83833640666933, −10.71803050375839, −10.14009929544849, −9.523163014361230, −9.118424892767970, −8.589475725278435, −8.047717552254978, −7.924461339827246, −6.902085292897392, −6.438938350053229, −6.325441307366974, −5.634332051345798, −4.943557612847104, −4.166686284058753, −3.799854925865220, −2.982273865591414, −2.585986297589011, −1.865875197080452, −1.050530603859885, −0.4678359957439603,
0.4678359957439603, 1.050530603859885, 1.865875197080452, 2.585986297589011, 2.982273865591414, 3.799854925865220, 4.166686284058753, 4.943557612847104, 5.634332051345798, 6.325441307366974, 6.438938350053229, 6.902085292897392, 7.924461339827246, 8.047717552254978, 8.589475725278435, 9.118424892767970, 9.523163014361230, 10.14009929544849, 10.71803050375839, 10.83833640666933, 11.36634870385757, 12.02995728916267, 12.74263201520404, 12.84541353063839, 13.33471643646272
