| L(s) = 1 | + 5-s − 3·7-s − 5·11-s − 5·17-s − 19-s − 23-s + 25-s + 9·29-s − 4·31-s − 3·35-s + 3·37-s − 41-s + 3·43-s + 8·47-s + 2·49-s − 10·53-s − 5·55-s − 3·59-s + 7·61-s − 9·67-s − 7·71-s − 10·73-s + 15·77-s + 16·79-s − 12·83-s − 5·85-s + 15·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.21·17-s − 0.229·19-s − 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.718·31-s − 0.507·35-s + 0.493·37-s − 0.156·41-s + 0.457·43-s + 1.16·47-s + 2/7·49-s − 1.37·53-s − 0.674·55-s − 0.390·59-s + 0.896·61-s − 1.09·67-s − 0.830·71-s − 1.17·73-s + 1.70·77-s + 1.80·79-s − 1.31·83-s − 0.542·85-s + 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.56356852443007, −13.29207096297523, −12.96094103276567, −12.44630780521186, −12.02204954728258, −11.26489937474618, −10.72194564123815, −10.42503832334946, −9.971403218221361, −9.433476789153681, −8.898832454904328, −8.531718154872987, −7.779493473442317, −7.407584771752742, −6.703272330522870, −6.282408119984158, −5.902083422841506, −5.198914021828106, −4.690238894795008, −4.157539324569604, −3.343999227407690, −2.768760625857585, −2.448347523718104, −1.707954962908037, −0.6603269919762897, 0,
0.6603269919762897, 1.707954962908037, 2.448347523718104, 2.768760625857585, 3.343999227407690, 4.157539324569604, 4.690238894795008, 5.198914021828106, 5.902083422841506, 6.282408119984158, 6.703272330522870, 7.407584771752742, 7.779493473442317, 8.531718154872987, 8.898832454904328, 9.433476789153681, 9.971403218221361, 10.42503832334946, 10.72194564123815, 11.26489937474618, 12.02204954728258, 12.44630780521186, 12.96094103276567, 13.29207096297523, 13.56356852443007
