| L(s) = 1 | + 3-s − 5-s + 9-s − 2·13-s − 15-s + 6·17-s − 8·19-s + 25-s + 27-s + 2·29-s − 6·37-s − 2·39-s + 41-s + 12·43-s − 45-s − 7·49-s + 6·51-s − 2·53-s − 8·57-s − 4·59-s + 2·61-s + 2·65-s − 4·67-s − 12·71-s + 10·73-s + 75-s + 4·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.986·37-s − 0.320·39-s + 0.156·41-s + 1.82·43-s − 0.149·45-s − 49-s + 0.840·51-s − 0.274·53-s − 1.05·57-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.115·75-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.93439847741864, −14.52627721639323, −14.22778288045896, −13.55846979184741, −12.86366650683508, −12.45479984909532, −12.15456637370293, −11.40711292732866, −10.75781785129342, −10.32074726412237, −9.841379246661107, −9.059731923778446, −8.769174344369065, −7.921489694192138, −7.779499406845391, −7.068144377627827, −6.413377656577732, −5.857527028885692, −5.022220082102674, −4.512695215959930, −3.839908349800216, −3.287032209712129, −2.566578508381694, −1.920270839526811, −1.016097642819346, 0,
1.016097642819346, 1.920270839526811, 2.566578508381694, 3.287032209712129, 3.839908349800216, 4.512695215959930, 5.022220082102674, 5.857527028885692, 6.413377656577732, 7.068144377627827, 7.779499406845391, 7.921489694192138, 8.769174344369065, 9.059731923778446, 9.841379246661107, 10.32074726412237, 10.75781785129342, 11.40711292732866, 12.15456637370293, 12.45479984909532, 12.86366650683508, 13.55846979184741, 14.22778288045896, 14.52627721639323, 14.93439847741864
