| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s − 6·13-s + 14-s + 16-s − 17-s + 2·20-s + 8·23-s − 25-s + 6·26-s − 28-s + 6·29-s − 8·31-s − 32-s + 34-s − 2·35-s + 10·37-s − 2·40-s + 6·41-s + 12·43-s − 8·46-s + 49-s + 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.447·20-s + 1.66·23-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.171·34-s − 0.338·35-s + 1.64·37-s − 0.316·40-s + 0.937·41-s + 1.82·43-s − 1.17·46-s + 1/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.306032340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.306032340\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.352367073730369250319535139199, −8.466864404810002345434086097975, −7.34947934570743894088685704902, −7.02175028110277092253993140442, −5.96218325167265543606565738451, −5.29944243683183488334817629056, −4.24467666749940526398113891006, −2.78295861731664811401382989703, −2.27199331297833251963893517833, −0.824579001110863552316391184137,
0.824579001110863552316391184137, 2.27199331297833251963893517833, 2.78295861731664811401382989703, 4.24467666749940526398113891006, 5.29944243683183488334817629056, 5.96218325167265543606565738451, 7.02175028110277092253993140442, 7.34947934570743894088685704902, 8.466864404810002345434086097975, 9.352367073730369250319535139199
