| L(s) = 1 | + 3-s + 5-s + 2·7-s + 9-s − 6·13-s + 15-s + 2·21-s + 2·23-s + 25-s + 27-s − 4·29-s − 8·31-s + 2·35-s + 6·37-s − 6·39-s − 6·41-s + 8·43-s + 45-s + 6·47-s − 3·49-s + 2·53-s + 2·59-s − 4·61-s + 2·63-s − 6·65-s + 67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.66·13-s + 0.258·15-s + 0.436·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.960·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.274·53-s + 0.260·59-s − 0.512·61-s + 0.251·63-s − 0.744·65-s + 0.122·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 67 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.62439209881234, −13.98003271754432, −13.63921907158009, −12.92787905331202, −12.53145684783631, −12.11771057997393, −11.35561282562741, −10.96663528547629, −10.40312787998771, −9.766104684403869, −9.349226778473218, −9.014192105027111, −8.232777830760527, −7.749496823638301, −7.307893335199278, −6.854704301024472, −6.056745954782361, −5.279594017171127, −5.096652508856375, −4.310110655449071, −3.777870147911197, −2.912600218802302, −2.350407878998020, −1.890990500614373, −1.078791751836463, 0,
1.078791751836463, 1.890990500614373, 2.350407878998020, 2.912600218802302, 3.777870147911197, 4.310110655449071, 5.096652508856375, 5.279594017171127, 6.056745954782361, 6.854704301024472, 7.307893335199278, 7.749496823638301, 8.232777830760527, 9.014192105027111, 9.349226778473218, 9.766104684403869, 10.40312787998771, 10.96663528547629, 11.35561282562741, 12.11771057997393, 12.53145684783631, 12.92787905331202, 13.63921907158009, 13.98003271754432, 14.62439209881234
