| L(s) = 1 | − 3-s − 5-s + 9-s + 6·11-s + 2·13-s + 15-s + 4·19-s + 2·23-s + 25-s − 27-s + 4·29-s + 8·31-s − 6·33-s − 2·37-s − 2·39-s + 8·41-s − 4·43-s − 45-s − 4·47-s − 7·49-s + 8·53-s − 6·55-s − 4·57-s − 12·59-s + 10·61-s − 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.80·11-s + 0.554·13-s + 0.258·15-s + 0.917·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 1.43·31-s − 1.04·33-s − 0.328·37-s − 0.320·39-s + 1.24·41-s − 0.609·43-s − 0.149·45-s − 0.583·47-s − 49-s + 1.09·53-s − 0.809·55-s − 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.634168818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.634168818\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 79 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.79993503596502, −14.33583503951993, −13.84869235097884, −13.33718824985843, −12.57346527987975, −12.12282578630879, −11.71713221491512, −11.25029752846588, −10.86479041007043, −9.917988533338365, −9.707989050702846, −8.943344266659300, −8.483675341939962, −7.853323673233385, −7.136701453927626, −6.583660398866993, −6.286331643003220, −5.539741970654493, −4.759130961089927, −4.349684265603249, −3.579604675829665, −3.166725143521172, −2.058048375460771, −1.121141792281798, −0.7728445768713942,
0.7728445768713942, 1.121141792281798, 2.058048375460771, 3.166725143521172, 3.579604675829665, 4.349684265603249, 4.759130961089927, 5.539741970654493, 6.286331643003220, 6.583660398866993, 7.136701453927626, 7.853323673233385, 8.483675341939962, 8.943344266659300, 9.707989050702846, 9.917988533338365, 10.86479041007043, 11.25029752846588, 11.71713221491512, 12.12282578630879, 12.57346527987975, 13.33718824985843, 13.84869235097884, 14.33583503951993, 14.79993503596502
