| L(s) = 1 | − 2-s − 4-s + 2·7-s + 3·8-s + 2·11-s + 13-s − 2·14-s − 16-s + 17-s − 2·22-s − 8·23-s − 26-s − 2·28-s + 6·29-s + 6·31-s − 5·32-s − 34-s − 4·37-s + 12·41-s + 4·43-s − 2·44-s + 8·46-s − 3·49-s − 52-s − 6·53-s + 6·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 1.06·8-s + 0.603·11-s + 0.277·13-s − 0.534·14-s − 1/4·16-s + 0.242·17-s − 0.426·22-s − 1.66·23-s − 0.196·26-s − 0.377·28-s + 1.11·29-s + 1.07·31-s − 0.883·32-s − 0.171·34-s − 0.657·37-s + 1.87·41-s + 0.609·43-s − 0.301·44-s + 1.17·46-s − 3/7·49-s − 0.138·52-s − 0.824·53-s + 0.801·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593905683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593905683\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−14.35962152806353, −13.93501819117744, −13.86314315625809, −12.92595266352607, −12.46904245163637, −11.85631965605343, −11.45693651358603, −10.73912780837563, −10.33293196551706, −9.842931326040873, −9.239880978622464, −8.814936959070219, −8.171604196134532, −7.879025698566919, −7.390382189794641, −6.445251911767253, −6.106819473762842, −5.281278900624305, −4.650400319292491, −4.222289522331916, −3.649153801543428, −2.698697528319882, −1.871620731536229, −1.253539742941542, −0.5631946575207538,
0.5631946575207538, 1.253539742941542, 1.871620731536229, 2.698697528319882, 3.649153801543428, 4.222289522331916, 4.650400319292491, 5.281278900624305, 6.106819473762842, 6.445251911767253, 7.390382189794641, 7.879025698566919, 8.171604196134532, 8.814936959070219, 9.239880978622464, 9.842931326040873, 10.33293196551706, 10.73912780837563, 11.45693651358603, 11.85631965605343, 12.46904245163637, 12.92595266352607, 13.86314315625809, 13.93501819117744, 14.35962152806353
