| L(s) = 1 | − 5-s + 11-s − 13-s + 4·17-s + 3·19-s + 6·23-s − 4·25-s − 7·29-s + 4·31-s + 37-s − 4·41-s + 2·43-s − 7·47-s − 10·53-s − 55-s + 9·59-s + 2·61-s + 65-s + 9·67-s − 4·71-s + 9·73-s − 16·79-s − 4·83-s − 4·85-s − 2·89-s − 3·95-s + 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.301·11-s − 0.277·13-s + 0.970·17-s + 0.688·19-s + 1.25·23-s − 4/5·25-s − 1.29·29-s + 0.718·31-s + 0.164·37-s − 0.624·41-s + 0.304·43-s − 1.02·47-s − 1.37·53-s − 0.134·55-s + 1.17·59-s + 0.256·61-s + 0.124·65-s + 1.09·67-s − 0.474·71-s + 1.05·73-s − 1.80·79-s − 0.439·83-s − 0.433·85-s − 0.211·89-s − 0.307·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−14.46793751884002, −13.76890836539073, −13.24431445374091, −12.77691418097685, −12.29889011608345, −11.63085128057961, −11.44195024505593, −10.92770930196410, −10.10200239175830, −9.791699429215668, −9.316406507308873, −8.696344023860606, −8.064126765509506, −7.697176657934939, −7.130024515602705, −6.667437461445097, −5.926272101010128, −5.384345399293655, −4.916623495937916, −4.224578999008494, −3.535540768158128, −3.202767653520415, −2.404201513459703, −1.567156650449412, −0.9342330256512383, 0,
0.9342330256512383, 1.567156650449412, 2.404201513459703, 3.202767653520415, 3.535540768158128, 4.224578999008494, 4.916623495937916, 5.384345399293655, 5.926272101010128, 6.667437461445097, 7.130024515602705, 7.697176657934939, 8.064126765509506, 8.696344023860606, 9.316406507308873, 9.791699429215668, 10.10200239175830, 10.92770930196410, 11.44195024505593, 11.63085128057961, 12.29889011608345, 12.77691418097685, 13.24431445374091, 13.76890836539073, 14.46793751884002
