| L(s) = 1 | + 3-s − 4·7-s + 9-s + 5·13-s + 6·17-s − 19-s − 4·21-s − 4·23-s + 27-s − 2·29-s − 5·31-s + 11·37-s + 5·39-s + 9·43-s − 2·47-s + 9·49-s + 6·51-s − 6·53-s − 57-s − 10·59-s − 15·61-s − 4·63-s + 3·67-s − 4·69-s − 10·71-s + 7·73-s + 13·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.38·13-s + 1.45·17-s − 0.229·19-s − 0.872·21-s − 0.834·23-s + 0.192·27-s − 0.371·29-s − 0.898·31-s + 1.80·37-s + 0.800·39-s + 1.37·43-s − 0.291·47-s + 9/7·49-s + 0.840·51-s − 0.824·53-s − 0.132·57-s − 1.30·59-s − 1.92·61-s − 0.503·63-s + 0.366·67-s − 0.481·69-s − 1.18·71-s + 0.819·73-s + 1.46·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 15 T + p T^{2} \) | 1.61.p |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−14.15350020869262, −13.95542973967787, −13.23477413960772, −12.98522657854962, −12.40706472271812, −12.10361557190882, −11.20958999422897, −10.80956668131825, −10.24345253292412, −9.680762530399025, −9.288234070926835, −8.971520800235750, −8.077738323786073, −7.798076056259970, −7.246381814635888, −6.422907693608455, −6.021077620583553, −5.800306386854160, −4.791527744276128, −3.983811530085699, −3.650956517523068, −3.110972484416620, −2.580435313684257, −1.636707968737303, −0.9783324810352119, 0,
0.9783324810352119, 1.636707968737303, 2.580435313684257, 3.110972484416620, 3.650956517523068, 3.983811530085699, 4.791527744276128, 5.800306386854160, 6.021077620583553, 6.422907693608455, 7.246381814635888, 7.798076056259970, 8.077738323786073, 8.971520800235750, 9.288234070926835, 9.680762530399025, 10.24345253292412, 10.80956668131825, 11.20958999422897, 12.10361557190882, 12.40706472271812, 12.98522657854962, 13.23477413960772, 13.95542973967787, 14.15350020869262
