| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 13-s − 14-s − 2·15-s + 16-s + 6·17-s − 18-s + 4·19-s − 2·20-s + 21-s + 8·23-s − 24-s − 25-s + 26-s + 27-s + 28-s + 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-s + 1.66·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.088072763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.088072763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 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.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.35333426053740, −13.86589494330645, −13.08694197792099, −12.59246857382238, −12.20397610512677, −11.50189338663260, −11.23013823734398, −10.70875490678257, −9.983110277329365, −9.500335124121889, −9.205738303596371, −8.408840724914499, −8.040099870845386, −7.527718064286048, −7.295741987480355, −6.614185711416555, −5.780253881628626, −5.179220048575989, −4.650130215188058, −3.804310914018118, −3.250572681655687, −2.899835581596258, −1.920013291549964, −1.235915886837018, −0.5785458328995170,
0.5785458328995170, 1.235915886837018, 1.920013291549964, 2.899835581596258, 3.250572681655687, 3.804310914018118, 4.650130215188058, 5.179220048575989, 5.780253881628626, 6.614185711416555, 7.295741987480355, 7.527718064286048, 8.040099870845386, 8.408840724914499, 9.205738303596371, 9.500335124121889, 9.983110277329365, 10.70875490678257, 11.23013823734398, 11.50189338663260, 12.20397610512677, 12.59246857382238, 13.08694197792099, 13.86589494330645, 14.35333426053740
