| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s + 9-s − 3·10-s + 3·11-s − 12-s − 13-s + 3·15-s + 16-s − 6·17-s + 18-s + 4·19-s − 3·20-s + 3·22-s + 6·23-s − 24-s + 4·25-s − 26-s − 27-s − 9·29-s + 3·30-s − 5·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s − 0.288·12-s − 0.277·13-s + 0.774·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.670·20-s + 0.639·22-s + 1.25·23-s − 0.204·24-s + 4/5·25-s − 0.196·26-s − 0.192·27-s − 1.67·29-s + 0.547·30-s − 0.898·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 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.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−7.80449742215943975642408683950, −7.22215078412481617287398797495, −6.77329950857015176150472848829, −5.77063328373617077491226485820, −5.03579826378674901085655641329, −4.15410368415315111753139715356, −3.80832284502297766837948375653, −2.71940653419962020587810779241, −1.38318779336578517050705007355, 0,
1.38318779336578517050705007355, 2.71940653419962020587810779241, 3.80832284502297766837948375653, 4.15410368415315111753139715356, 5.03579826378674901085655641329, 5.77063328373617077491226485820, 6.77329950857015176150472848829, 7.22215078412481617287398797495, 7.80449742215943975642408683950
