| L(s) = 1 | + 2·3-s + 2·5-s + 9-s − 6·11-s − 2·13-s + 4·15-s − 17-s + 6·23-s − 25-s − 4·27-s − 10·29-s − 2·31-s − 12·33-s + 6·37-s − 4·39-s + 6·41-s − 8·43-s + 2·45-s − 2·51-s − 10·53-s − 12·55-s + 8·59-s − 14·61-s − 4·65-s + 4·67-s + 12·69-s + 2·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 1.03·15-s − 0.242·17-s + 1.25·23-s − 1/5·25-s − 0.769·27-s − 1.85·29-s − 0.359·31-s − 2.08·33-s + 0.986·37-s − 0.640·39-s + 0.937·41-s − 1.21·43-s + 0.298·45-s − 0.280·51-s − 1.37·53-s − 1.61·55-s + 1.04·59-s − 1.79·61-s − 0.496·65-s + 0.488·67-s + 1.44·69-s + 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−7.71743605773314064060517511453, −7.22320038345813934406313365840, −6.15587167996465714283741394082, −5.41901376085817111414682486932, −4.92852786911030460828356623060, −3.80276371358035318939394169571, −2.88205489232261764953541720461, −2.46005267455719907629170520175, −1.69352370897343893726020061854, 0,
1.69352370897343893726020061854, 2.46005267455719907629170520175, 2.88205489232261764953541720461, 3.80276371358035318939394169571, 4.92852786911030460828356623060, 5.41901376085817111414682486932, 6.15587167996465714283741394082, 7.22320038345813934406313365840, 7.71743605773314064060517511453
