| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s + 15-s + 2·17-s − 2·21-s + 25-s − 27-s − 8·29-s − 31-s − 2·35-s − 8·37-s + 2·41-s − 45-s + 8·47-s − 3·49-s − 2·51-s + 6·53-s − 10·59-s − 6·61-s + 2·63-s − 14·67-s + 6·71-s − 4·73-s − 75-s + 16·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.258·15-s + 0.485·17-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s − 0.338·35-s − 1.31·37-s + 0.312·41-s − 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 1.30·59-s − 0.768·61-s + 0.251·63-s − 1.71·67-s + 0.712·71-s − 0.468·73-s − 0.115·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 31 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.56526130541601986938794648304, −6.96388089713072950217738841251, −6.04472390692213196680642619858, −5.41991814742730965032081243626, −4.78125618235390295963542640944, −4.01363798010063988694991193625, −3.28057075746859495065584782278, −2.09055683176754705208009690419, −1.23129174912836715450483203121, 0,
1.23129174912836715450483203121, 2.09055683176754705208009690419, 3.28057075746859495065584782278, 4.01363798010063988694991193625, 4.78125618235390295963542640944, 5.41991814742730965032081243626, 6.04472390692213196680642619858, 6.96388089713072950217738841251, 7.56526130541601986938794648304
