| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s − 3·11-s + 3·15-s − 7·17-s + 21-s − 4·23-s + 4·25-s + 27-s − 4·29-s − 4·31-s − 3·33-s + 3·35-s + 4·37-s − 12·41-s + 43-s + 3·45-s − 9·47-s − 6·49-s − 7·51-s − 12·53-s − 9·55-s − 8·59-s − 5·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.774·15-s − 1.69·17-s + 0.218·21-s − 0.834·23-s + 4/5·25-s + 0.192·27-s − 0.742·29-s − 0.718·31-s − 0.522·33-s + 0.507·35-s + 0.657·37-s − 1.87·41-s + 0.152·43-s + 0.447·45-s − 1.31·47-s − 6/7·49-s − 0.980·51-s − 1.64·53-s − 1.21·55-s − 1.04·59-s − 0.640·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.59698636069540757484949403922, −6.48420023253247087955657068999, −6.29115188930501012140901417658, −5.15022714585569524482302464698, −4.87128994588008274336549820951, −3.79884039990866780165028982928, −2.89320691013502015523993091285, −1.99340799726519623573848293305, −1.77557215543104849151652825273, 0,
1.77557215543104849151652825273, 1.99340799726519623573848293305, 2.89320691013502015523993091285, 3.79884039990866780165028982928, 4.87128994588008274336549820951, 5.15022714585569524482302464698, 6.29115188930501012140901417658, 6.48420023253247087955657068999, 7.59698636069540757484949403922
