| L(s) = 1 | + 2·2-s + 2·4-s + 6·11-s − 4·13-s − 4·16-s − 7·17-s + 5·19-s + 12·22-s + 23-s − 5·25-s − 8·26-s − 29-s + 4·31-s − 8·32-s − 14·34-s + 3·37-s + 10·38-s − 6·43-s + 12·44-s + 2·46-s + 9·47-s − 10·50-s − 8·52-s − 10·53-s − 2·58-s + 3·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.80·11-s − 1.10·13-s − 16-s − 1.69·17-s + 1.14·19-s + 2.55·22-s + 0.208·23-s − 25-s − 1.56·26-s − 0.185·29-s + 0.718·31-s − 1.41·32-s − 2.40·34-s + 0.493·37-s + 1.62·38-s − 0.914·43-s + 1.80·44-s + 0.294·46-s + 1.31·47-s − 1.41·50-s − 1.10·52-s − 1.37·53-s − 0.262·58-s + 0.390·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−15.19564328018777, −14.90516779455100, −14.14336638820784, −13.87723368121745, −13.52929377148227, −12.71254385923713, −12.30784251734991, −11.82009757226769, −11.36373076050983, −11.00826602077138, −9.870714068501117, −9.494359534061597, −9.057292689050081, −8.329065638952261, −7.466328354365717, −6.874626517542897, −6.459976069366051, −5.941971626332142, −5.079496806103854, −4.729687295830739, −3.969396641194194, −3.695242489200129, −2.720992893797571, −2.199616191492174, −1.249445811929363, 0,
1.249445811929363, 2.199616191492174, 2.720992893797571, 3.695242489200129, 3.969396641194194, 4.729687295830739, 5.079496806103854, 5.941971626332142, 6.459976069366051, 6.874626517542897, 7.466328354365717, 8.329065638952261, 9.057292689050081, 9.494359534061597, 9.870714068501117, 11.00826602077138, 11.36373076050983, 11.82009757226769, 12.30784251734991, 12.71254385923713, 13.52929377148227, 13.87723368121745, 14.14336638820784, 14.90516779455100, 15.19564328018777
