| L(s) = 1 | + 2·5-s + 4·7-s + 2·13-s − 2·17-s − 4·19-s − 25-s − 2·29-s − 8·31-s + 8·35-s + 2·37-s − 10·41-s − 4·43-s + 9·49-s − 6·53-s + 12·59-s + 2·61-s + 4·65-s − 12·67-s + 16·71-s + 10·73-s + 4·79-s − 4·85-s + 6·89-s + 8·91-s − 8·95-s + 14·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 1.35·35-s + 0.328·37-s − 1.56·41-s − 0.609·43-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-s + 1.89·71-s + 1.17·73-s + 0.450·79-s − 0.433·85-s + 0.635·89-s + 0.838·91-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.05731554975351, −12.51758109566200, −11.86983892559954, −11.40947774173525, −11.11790029447104, −10.69687030720867, −10.16906213813692, −9.779877405467161, −9.090215542147791, −8.746050221360971, −8.397731535024693, −7.780762157557159, −7.468202756662098, −6.654381653617741, −6.405706055558500, −5.789846219548580, −5.219240975696137, −4.996864613642424, −4.364909927607632, −3.762550176431003, −3.332740055493610, −2.261994494403373, −2.043180707736972, −1.650440945581202, −0.9239268468007697, 0,
0.9239268468007697, 1.650440945581202, 2.043180707736972, 2.261994494403373, 3.332740055493610, 3.762550176431003, 4.364909927607632, 4.996864613642424, 5.219240975696137, 5.789846219548580, 6.405706055558500, 6.654381653617741, 7.468202756662098, 7.780762157557159, 8.397731535024693, 8.746050221360971, 9.090215542147791, 9.779877405467161, 10.16906213813692, 10.69687030720867, 11.11790029447104, 11.40947774173525, 11.86983892559954, 12.51758109566200, 13.05731554975351
