| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 12-s − 13-s − 16-s + 4·17-s + 18-s + 8·19-s + 3·24-s − 5·25-s − 26-s − 27-s − 4·29-s − 6·31-s + 5·32-s + 4·34-s − 36-s − 6·37-s + 8·38-s + 39-s − 6·41-s + 2·43-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.612·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.742·29-s − 1.07·31-s + 0.883·32-s + 0.685·34-s − 1/6·36-s − 0.986·37-s + 1.29·38-s + 0.160·39-s − 0.937·41-s + 0.304·43-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.543531729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.543531729\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−8.173661905691154395658256892118, −7.50509992116450690218486504904, −6.74441883637837806500684699064, −5.74292840263200101500545611439, −5.36129373510958848192499713524, −4.81293073414353286078140007434, −3.60177056872281193473735297123, −3.40501355132579861099004933610, −1.91854091282303381244006049252, −0.63821706936901217057862016536,
0.63821706936901217057862016536, 1.91854091282303381244006049252, 3.40501355132579861099004933610, 3.60177056872281193473735297123, 4.81293073414353286078140007434, 5.36129373510958848192499713524, 5.74292840263200101500545611439, 6.74441883637837806500684699064, 7.50509992116450690218486504904, 8.173661905691154395658256892118
