| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 7-s + 3·8-s + 9-s + 12-s + 13-s + 14-s − 16-s − 18-s + 8·19-s + 21-s + 8·23-s − 3·24-s − 5·25-s − 26-s − 27-s + 28-s − 4·29-s + 2·31-s − 5·32-s − 36-s + 10·37-s − 8·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.218·21-s + 1.66·23-s − 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.742·29-s + 0.359·31-s − 0.883·32-s − 1/6·36-s + 1.64·37-s − 1.29·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.36607862611142, −14.82549651688671, −14.16581280096249, −13.55911188542213, −13.14739884892312, −12.82790723332579, −11.96260571014715, −11.29838321137029, −11.24927540369519, −10.23049708900815, −9.946155267063388, −9.341413045895989, −9.078886111060573, −8.142035753255600, −7.802497970151757, −7.115968265533193, −6.644780110049352, −5.798599206904169, −5.226901633398740, −4.835390244588559, −3.922515591980382, −3.430447249269336, −2.543977155325327, −1.406021289043041, −0.9425443428662897, 0,
0.9425443428662897, 1.406021289043041, 2.543977155325327, 3.430447249269336, 3.922515591980382, 4.835390244588559, 5.226901633398740, 5.798599206904169, 6.644780110049352, 7.115968265533193, 7.802497970151757, 8.142035753255600, 9.078886111060573, 9.341413045895989, 9.946155267063388, 10.23049708900815, 11.24927540369519, 11.29838321137029, 11.96260571014715, 12.82790723332579, 13.14739884892312, 13.55911188542213, 14.16581280096249, 14.82549651688671, 15.36607862611142
