| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 3·9-s + 3·11-s − 3·14-s + 16-s + 17-s − 3·18-s + 6·19-s + 3·22-s + 3·23-s − 3·28-s − 6·29-s + 10·31-s + 32-s + 34-s − 3·36-s + 4·37-s + 6·38-s + 4·41-s − 11·43-s + 3·44-s + 3·46-s − 12·47-s + 2·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 9-s + 0.904·11-s − 0.801·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s + 1.37·19-s + 0.639·22-s + 0.625·23-s − 0.566·28-s − 1.11·29-s + 1.79·31-s + 0.176·32-s + 0.171·34-s − 1/2·36-s + 0.657·37-s + 0.973·38-s + 0.624·41-s − 1.67·43-s + 0.452·44-s + 0.442·46-s − 1.75·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.49582985451830, −13.26011704001897, −12.76908804907430, −12.08628022787943, −11.82684089512804, −11.28292098152506, −11.09652440131715, −10.03171560724000, −9.886667246123755, −9.357885388645670, −8.873211406398587, −8.160359472211540, −7.807072131010511, −6.945474474681702, −6.741666746048597, −6.047640809516881, −5.867085029595981, −5.052819493146583, −4.725057562894724, −3.864884236706414, −3.278419679085419, −3.138709228358100, −2.479280085077428, −1.564769260080551, −0.9084019326805731, 0,
0.9084019326805731, 1.564769260080551, 2.479280085077428, 3.138709228358100, 3.278419679085419, 3.864884236706414, 4.725057562894724, 5.052819493146583, 5.867085029595981, 6.047640809516881, 6.741666746048597, 6.945474474681702, 7.807072131010511, 8.160359472211540, 8.873211406398587, 9.357885388645670, 9.886667246123755, 10.03171560724000, 11.09652440131715, 11.28292098152506, 11.82684089512804, 12.08628022787943, 12.76908804907430, 13.26011704001897, 13.49582985451830
