| L(s) = 1 | − 2·3-s + 5-s − 2·7-s + 9-s − 2·13-s − 2·15-s − 6·17-s − 4·19-s + 4·21-s − 6·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s − 2·35-s − 2·37-s + 4·39-s + 6·41-s − 10·43-s + 45-s + 6·47-s − 3·49-s + 12·51-s + 6·53-s + 8·57-s + 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s + 0.640·39-s + 0.937·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 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.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−11.16454648418111686213251777526, −10.35239107507603235483123434521, −9.511328653906225471558818278712, −8.404829898294941223649273971532, −6.87772159023272676690262517558, −6.26903409609752801893873868553, −5.32703134203753576211066117067, −4.13045861856120535222472022668, −2.32736297473362010207341859714, 0,
2.32736297473362010207341859714, 4.13045861856120535222472022668, 5.32703134203753576211066117067, 6.26903409609752801893873868553, 6.87772159023272676690262517558, 8.404829898294941223649273971532, 9.511328653906225471558818278712, 10.35239107507603235483123434521, 11.16454648418111686213251777526
