| L(s) = 1 | + 3-s − 5-s + 9-s − 2·11-s − 4·13-s − 15-s + 25-s + 27-s − 2·29-s − 10·31-s − 2·33-s − 4·37-s − 4·39-s + 6·41-s − 4·43-s − 45-s − 7·49-s − 2·53-s + 2·55-s − 6·59-s − 6·61-s + 4·65-s − 4·67-s − 8·71-s + 14·73-s + 75-s − 2·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.79·31-s − 0.348·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 49-s − 0.274·53-s + 0.269·55-s − 0.781·59-s − 0.768·61-s + 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.63·73-s + 0.115·75-s − 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−8.889712831101859103804444847531, −7.80559821824428200433846976691, −7.54557047142823067823597904482, −6.60385394707253465783797985484, −5.43724899639797774658826630136, −4.70481521934788449769197313811, −3.70081218601629080212569355255, −2.83139366584180811615185959512, −1.81084238627342889349146058738, 0,
1.81084238627342889349146058738, 2.83139366584180811615185959512, 3.70081218601629080212569355255, 4.70481521934788449769197313811, 5.43724899639797774658826630136, 6.60385394707253465783797985484, 7.54557047142823067823597904482, 7.80559821824428200433846976691, 8.889712831101859103804444847531
