| L(s) = 1 | − 2·7-s − 2·11-s + 2·13-s − 6·17-s + 8·19-s + 4·23-s − 8·29-s − 10·37-s − 2·41-s − 12·43-s − 3·49-s − 10·53-s + 6·59-s + 2·61-s − 8·67-s + 4·71-s + 4·73-s + 4·77-s − 8·79-s − 4·83-s − 6·89-s − 4·91-s + 8·97-s − 2·103-s − 4·107-s − 6·109-s − 2·113-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.603·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 0.834·23-s − 1.48·29-s − 1.64·37-s − 0.312·41-s − 1.82·43-s − 3/7·49-s − 1.37·53-s + 0.781·59-s + 0.256·61-s − 0.977·67-s + 0.474·71-s + 0.468·73-s + 0.455·77-s − 0.900·79-s − 0.439·83-s − 0.635·89-s − 0.419·91-s + 0.812·97-s − 0.197·103-s − 0.386·107-s − 0.574·109-s − 0.188·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.988892191864512458013193817471, −8.136056360422255191066760940012, −7.15562382903539764269849136934, −6.63560342158760211146375054979, −5.56718290716607818845063804153, −4.90786055385638384453526942829, −3.63702338059271712194862462334, −2.98769726639906032012813910091, −1.65944442938371927957526732518, 0,
1.65944442938371927957526732518, 2.98769726639906032012813910091, 3.63702338059271712194862462334, 4.90786055385638384453526942829, 5.56718290716607818845063804153, 6.63560342158760211146375054979, 7.15562382903539764269849136934, 8.136056360422255191066760940012, 8.988892191864512458013193817471
