| L(s) = 1 | − 3-s − 7-s + 9-s + 3·11-s − 2·13-s + 17-s − 4·19-s + 21-s + 9·23-s − 27-s + 6·29-s − 7·31-s − 3·33-s + 4·37-s + 2·39-s − 6·41-s + 4·43-s − 3·47-s + 49-s − 51-s − 12·53-s + 4·57-s − 3·59-s − 61-s − 63-s − 5·67-s − 9·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s + 0.218·21-s + 1.87·23-s − 0.192·27-s + 1.11·29-s − 1.25·31-s − 0.522·33-s + 0.657·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.437·47-s + 1/7·49-s − 0.140·51-s − 1.64·53-s + 0.529·57-s − 0.390·59-s − 0.128·61-s − 0.125·63-s − 0.610·67-s − 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.23360787119393, −14.51646636425771, −14.40207441984272, −13.48977228504803, −12.98961651522742, −12.49610612575618, −12.16340551077560, −11.42592733681796, −10.96550962607559, −10.58185458112117, −9.744313168344331, −9.443531762254715, −8.824482633343530, −8.240883289479308, −7.435354972550483, −6.920403601231555, −6.470881278923465, −5.954904536585004, −5.102585084972413, −4.738554388194141, −3.997773697940910, −3.303624791393954, −2.634679082747410, −1.687705305508358, −0.9651605169435147, 0,
0.9651605169435147, 1.687705305508358, 2.634679082747410, 3.303624791393954, 3.997773697940910, 4.738554388194141, 5.102585084972413, 5.954904536585004, 6.470881278923465, 6.920403601231555, 7.435354972550483, 8.240883289479308, 8.824482633343530, 9.443531762254715, 9.744313168344331, 10.58185458112117, 10.96550962607559, 11.42592733681796, 12.16340551077560, 12.49610612575618, 12.98961651522742, 13.48977228504803, 14.40207441984272, 14.51646636425771, 15.23360787119393
