| L(s) = 1 | + 3-s − 5-s + 9-s − 4·13-s − 15-s − 4·17-s − 4·19-s + 8·23-s + 25-s + 27-s + 2·29-s + 2·31-s − 6·37-s − 4·39-s + 6·41-s − 45-s + 12·47-s − 7·49-s − 4·51-s + 2·53-s − 4·57-s + 6·59-s − 14·61-s + 4·65-s + 67-s + 8·69-s + 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.10·13-s − 0.258·15-s − 0.970·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.359·31-s − 0.986·37-s − 0.640·39-s + 0.937·41-s − 0.149·45-s + 1.75·47-s − 49-s − 0.560·51-s + 0.274·53-s − 0.529·57-s + 0.781·59-s − 1.79·61-s + 0.496·65-s + 0.122·67-s + 0.963·69-s + 1.66·71-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 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 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
−14.53367613364882, −13.99323513552379, −13.55347480906443, −12.85786060648700, −12.57335952003957, −12.11853344479276, −11.37513570930021, −10.93006310624990, −10.47932804384658, −9.887026498917881, −9.164695882844617, −8.938720277220635, −8.371149148332324, −7.735496464483770, −7.265827733517547, −6.749517068845185, −6.314055591376283, −5.306083312860302, −4.890886889087289, −4.275858914018354, −3.813702308453880, −2.868279145453808, −2.594274907437554, −1.836813382359690, −0.8957179675146910, 0,
0.8957179675146910, 1.836813382359690, 2.594274907437554, 2.868279145453808, 3.813702308453880, 4.275858914018354, 4.890886889087289, 5.306083312860302, 6.314055591376283, 6.749517068845185, 7.265827733517547, 7.735496464483770, 8.371149148332324, 8.938720277220635, 9.164695882844617, 9.887026498917881, 10.47932804384658, 10.93006310624990, 11.37513570930021, 12.11853344479276, 12.57335952003957, 12.85786060648700, 13.55347480906443, 13.99323513552379, 14.53367613364882
