| L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 4·13-s − 2·14-s + 16-s − 8·17-s − 8·19-s + 23-s − 5·25-s + 4·26-s − 2·28-s + 3·29-s + 32-s − 8·34-s + 7·37-s − 8·38-s − 12·41-s − 11·43-s + 46-s − 6·47-s − 3·49-s − 5·50-s + 4·52-s − 5·53-s − 2·56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.94·17-s − 1.83·19-s + 0.208·23-s − 25-s + 0.784·26-s − 0.377·28-s + 0.557·29-s + 0.176·32-s − 1.37·34-s + 1.15·37-s − 1.29·38-s − 1.87·41-s − 1.67·43-s + 0.147·46-s − 0.875·47-s − 3/7·49-s − 0.707·50-s + 0.554·52-s − 0.686·53-s − 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 172062 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 172062 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 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
−13.54903511611226, −13.15337709470288, −13.03234048079857, −12.33711532711968, −11.81995837687361, −11.31630288161228, −10.92276274124705, −10.50766202143086, −9.983306213429965, −9.376003098813942, −8.849445066234997, −8.377177126693979, −8.041211251862230, −7.197293923794194, −6.530311276322964, −6.369019333913136, −6.167850200190444, −5.258787411575130, −4.622884977502073, −4.275721197205799, −3.780255319246787, −3.064486759395066, −2.705314745948012, −1.725797398577206, −1.608795781240418, 0, 0,
1.608795781240418, 1.725797398577206, 2.705314745948012, 3.064486759395066, 3.780255319246787, 4.275721197205799, 4.622884977502073, 5.258787411575130, 6.167850200190444, 6.369019333913136, 6.530311276322964, 7.197293923794194, 8.041211251862230, 8.377177126693979, 8.849445066234997, 9.376003098813942, 9.983306213429965, 10.50766202143086, 10.92276274124705, 11.31630288161228, 11.81995837687361, 12.33711532711968, 13.03234048079857, 13.15337709470288, 13.54903511611226
