| L(s) = 1 | − 5-s − 2·11-s − 2·13-s − 6·17-s + 6·19-s + 25-s + 10·29-s + 8·31-s − 2·37-s + 6·41-s − 2·43-s − 12·47-s − 7·49-s + 10·53-s + 2·55-s + 6·59-s + 6·61-s + 2·65-s − 14·67-s − 4·71-s − 10·73-s − 8·79-s − 10·83-s + 6·85-s − 14·89-s − 6·95-s + 6·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.603·11-s − 0.554·13-s − 1.45·17-s + 1.37·19-s + 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.328·37-s + 0.937·41-s − 0.304·43-s − 1.75·47-s − 49-s + 1.37·53-s + 0.269·55-s + 0.781·59-s + 0.768·61-s + 0.248·65-s − 1.71·67-s − 0.474·71-s − 1.17·73-s − 0.900·79-s − 1.09·83-s + 0.650·85-s − 1.48·89-s − 0.615·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−7.81285724909768979483935524520, −7.01451501362389941870110582286, −6.51504632102335022204754268649, −5.51187964090898831671882097488, −4.75030050500533126512039347875, −4.26300454340832865134380080126, −3.03453794048438755656055087155, −2.58066945500031817459832592662, −1.24078453169086437489255286634, 0,
1.24078453169086437489255286634, 2.58066945500031817459832592662, 3.03453794048438755656055087155, 4.26300454340832865134380080126, 4.75030050500533126512039347875, 5.51187964090898831671882097488, 6.51504632102335022204754268649, 7.01451501362389941870110582286, 7.81285724909768979483935524520
