| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 8-s + 9-s + 2·10-s + 3·11-s + 12-s − 13-s − 2·15-s + 16-s + 5·17-s − 18-s + 19-s − 2·20-s − 3·22-s − 24-s − 25-s + 26-s + 27-s − 29-s + 2·30-s + 4·31-s − 32-s + 3·33-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 + 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.21·17-s − 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.639·22-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.185·29-s + 0.365·30-s + 0.718·31-s − 0.176·32-s + 0.522·33-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
| \(L(1)\) |
\(\approx\) |
\(1.480987321\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480987321\) |
| \(L(\frac{3}{2})\) |
|
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.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−8.297333116033476804495084625691, −7.957618144111480515946178088804, −7.24798617213873915158495705968, −6.56651905336329871997668256855, −5.60314673718101691299559177151, −4.51644889404065215441101905115, −3.67319303211476434636831552578, −3.04409235696481792419253753869, −1.84503885433940093290207535484, −0.78091454157602035310760967714,
0.78091454157602035310760967714, 1.84503885433940093290207535484, 3.04409235696481792419253753869, 3.67319303211476434636831552578, 4.51644889404065215441101905115, 5.60314673718101691299559177151, 6.56651905336329871997668256855, 7.24798617213873915158495705968, 7.957618144111480515946178088804, 8.297333116033476804495084625691
