| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 13-s − 15-s + 5·17-s − 8·19-s − 2·21-s + 6·23-s − 4·25-s + 27-s + 3·29-s + 4·31-s + 2·35-s − 11·37-s + 39-s − 3·41-s + 2·43-s − 45-s + 4·47-s − 3·49-s + 5·51-s + 3·53-s − 8·57-s + 6·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 1.21·17-s − 1.83·19-s − 0.436·21-s + 1.25·23-s − 4/5·25-s + 0.192·27-s + 0.557·29-s + 0.718·31-s + 0.338·35-s − 1.80·37-s + 0.160·39-s − 0.468·41-s + 0.304·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.700·51-s + 0.412·53-s − 1.05·57-s + 0.781·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−16.85296264014370, −15.88836811366471, −15.65907064519725, −14.98287942664303, −14.56731660676557, −13.74530517446717, −13.42024640219843, −12.52228129693231, −12.41747888165102, −11.59500121000688, −10.80843620769524, −10.27146269202786, −9.795974596738420, −8.923947487351360, −8.568881895912089, −7.920228174207252, −7.197355719398698, −6.621243408276639, −5.985198154439084, −5.113276287260233, −4.313397725838290, −3.612747169578744, −3.090972875021557, −2.233348740176525, −1.201170011812095, 0,
1.201170011812095, 2.233348740176525, 3.090972875021557, 3.612747169578744, 4.313397725838290, 5.113276287260233, 5.985198154439084, 6.621243408276639, 7.197355719398698, 7.920228174207252, 8.568881895912089, 8.923947487351360, 9.795974596738420, 10.27146269202786, 10.80843620769524, 11.59500121000688, 12.41747888165102, 12.52228129693231, 13.42024640219843, 13.74530517446717, 14.56731660676557, 14.98287942664303, 15.65907064519725, 15.88836811366471, 16.85296264014370
