| L(s) = 1 | + 2-s − 4-s − 2·7-s − 3·8-s + 2·11-s − 2·14-s − 16-s − 7·17-s − 8·19-s + 2·22-s − 2·23-s + 2·28-s + 3·29-s + 31-s + 5·32-s − 7·34-s + 8·37-s − 8·38-s − 12·41-s + 9·43-s − 2·44-s − 2·46-s + 13·47-s − 3·49-s − 6·53-s + 6·56-s + 3·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s + 0.603·11-s − 0.534·14-s − 1/4·16-s − 1.69·17-s − 1.83·19-s + 0.426·22-s − 0.417·23-s + 0.377·28-s + 0.557·29-s + 0.179·31-s + 0.883·32-s − 1.20·34-s + 1.31·37-s − 1.29·38-s − 1.87·41-s + 1.37·43-s − 0.301·44-s − 0.294·46-s + 1.89·47-s − 3/7·49-s − 0.824·53-s + 0.801·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.238445273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.238445273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−8.103413917217050421129466039245, −6.93131577517306742490877731732, −6.32529681012110759898973426369, −6.05309068067488172011544971874, −4.89266331527940472727178452581, −4.29094286474761853409390633151, −3.85265694089023747956553626434, −2.84200302581596003050899994355, −2.06901644965550948598940270696, −0.48574003301831647281953238142,
0.48574003301831647281953238142, 2.06901644965550948598940270696, 2.84200302581596003050899994355, 3.85265694089023747956553626434, 4.29094286474761853409390633151, 4.89266331527940472727178452581, 6.05309068067488172011544971874, 6.32529681012110759898973426369, 6.93131577517306742490877731732, 8.103413917217050421129466039245
