| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 4·11-s − 12-s − 13-s − 2·15-s + 16-s + 2·17-s − 18-s + 4·19-s + 2·20-s + 4·22-s − 4·23-s + 24-s − 25-s + 26-s − 27-s − 2·29-s + 2·30-s − 32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085289230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085289230\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.517682147336247724594873052480, −7.63527309524156674452712551160, −7.23648835012692702331338092518, −6.15799841254232847174211294216, −5.60568565401186827955815171542, −5.07459430002179485990417601973, −3.80628774523370370434092482714, −2.64593129595383993144676803692, −1.90464253431238082075610368258, −0.68078138016633670925521178705,
0.68078138016633670925521178705, 1.90464253431238082075610368258, 2.64593129595383993144676803692, 3.80628774523370370434092482714, 5.07459430002179485990417601973, 5.60568565401186827955815171542, 6.15799841254232847174211294216, 7.23648835012692702331338092518, 7.63527309524156674452712551160, 8.517682147336247724594873052480
