| L(s) = 1 | + 4·7-s + 5·13-s + 19-s − 5·25-s − 11·31-s − 37-s − 8·43-s + 9·49-s − 13·61-s + 16·67-s − 10·73-s + 4·79-s + 20·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.38·13-s + 0.229·19-s − 25-s − 1.97·31-s − 0.164·37-s − 1.21·43-s + 9/7·49-s − 1.66·61-s + 1.95·67-s − 1.17·73-s + 0.450·79-s + 2.09·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52272 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.66296757165770, −14.24607633126214, −13.72760926389849, −13.30166748904535, −12.74798449788156, −12.01040079971064, −11.58724752183588, −11.14543121285406, −10.74680762336200, −10.24588833042964, −9.346607965480364, −9.063981588282291, −8.214815849528926, −8.133853944205051, −7.436946201022332, −6.849157226047513, −6.113359220925427, −5.548293105711476, −5.125905848967819, −4.406178960845994, −3.798226461300201, −3.327578420027364, −2.265010109193704, −1.650347816378893, −1.209449718974947, 0,
1.209449718974947, 1.650347816378893, 2.265010109193704, 3.327578420027364, 3.798226461300201, 4.406178960845994, 5.125905848967819, 5.548293105711476, 6.113359220925427, 6.849157226047513, 7.436946201022332, 8.133853944205051, 8.214815849528926, 9.063981588282291, 9.346607965480364, 10.24588833042964, 10.74680762336200, 11.14543121285406, 11.58724752183588, 12.01040079971064, 12.74798449788156, 13.30166748904535, 13.72760926389849, 14.24607633126214, 14.66296757165770
