| L(s) = 1 | − 2-s + 4-s + 3·7-s − 8-s − 6·13-s − 3·14-s + 16-s + 7·17-s − 5·19-s − 6·23-s + 6·26-s + 3·28-s + 5·29-s − 3·31-s − 32-s − 7·34-s − 3·37-s + 5·38-s + 2·41-s + 4·43-s + 6·46-s − 2·47-s + 2·49-s − 6·52-s − 53-s − 3·56-s − 5·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.13·7-s − 0.353·8-s − 1.66·13-s − 0.801·14-s + 1/4·16-s + 1.69·17-s − 1.14·19-s − 1.25·23-s + 1.17·26-s + 0.566·28-s + 0.928·29-s − 0.538·31-s − 0.176·32-s − 1.20·34-s − 0.493·37-s + 0.811·38-s + 0.312·41-s + 0.609·43-s + 0.884·46-s − 0.291·47-s + 2/7·49-s − 0.832·52-s − 0.137·53-s − 0.400·56-s − 0.656·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−14.61304941955152, −14.45828180212698, −13.86583640827492, −13.03777271588751, −12.38046068227272, −11.96734410465430, −11.85086158262700, −10.85051225323881, −10.61967041024297, −9.891818989919220, −9.743054679016801, −8.883909909733598, −8.386232343343576, −7.787050994140126, −7.617185879313458, −6.957397763852799, −6.201351558564902, −5.642625295652558, −4.994683845901432, −4.533056773251094, −3.773698693586240, −2.957373951701172, −2.215629586316284, −1.792761997500540, −0.9155565065734313, 0,
0.9155565065734313, 1.792761997500540, 2.215629586316284, 2.957373951701172, 3.773698693586240, 4.533056773251094, 4.994683845901432, 5.642625295652558, 6.201351558564902, 6.957397763852799, 7.617185879313458, 7.787050994140126, 8.386232343343576, 8.883909909733598, 9.743054679016801, 9.891818989919220, 10.61967041024297, 10.85051225323881, 11.85086158262700, 11.96734410465430, 12.38046068227272, 13.03777271588751, 13.86583640827492, 14.45828180212698, 14.61304941955152
