| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 3·11-s − 12-s − 15-s + 16-s − 17-s − 18-s − 7·19-s + 20-s − 3·22-s + 2·23-s + 24-s + 25-s − 27-s − 7·29-s + 30-s − 10·31-s − 32-s − 3·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.60·19-s + 0.223·20-s − 0.639·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 1.29·29-s + 0.182·30-s − 1.79·31-s − 0.176·32-s − 0.522·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.94893443105744, −12.66474990554284, −12.13840166965922, −11.54052869081473, −11.04865349927968, −10.85956327933667, −10.46065661878738, −9.656005552666964, −9.420660445174945, −8.968215743880580, −8.618168472803528, −7.841029034818617, −7.455061296646733, −6.873784205334031, −6.529002664447785, −5.931744156862529, −5.679734569783695, −5.000911366964120, −4.215225508805060, −4.007672117110624, −3.236870771792858, −2.455637422655746, −1.913723272073394, −1.495850999910814, −0.6924160866569523, 0,
0.6924160866569523, 1.495850999910814, 1.913723272073394, 2.455637422655746, 3.236870771792858, 4.007672117110624, 4.215225508805060, 5.000911366964120, 5.679734569783695, 5.931744156862529, 6.529002664447785, 6.873784205334031, 7.455061296646733, 7.841029034818617, 8.618168472803528, 8.968215743880580, 9.420660445174945, 9.656005552666964, 10.46065661878738, 10.85956327933667, 11.04865349927968, 11.54052869081473, 12.13840166965922, 12.66474990554284, 12.94893443105744
