| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 4·11-s + 13-s + 16-s + 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s − 26-s + 6·29-s + 8·31-s − 32-s + 10·37-s − 4·38-s − 40-s − 6·41-s + 4·43-s + 4·44-s − 8·46-s − 7·49-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.277·13-s + 1/4·16-s + 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.196·26-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.967997593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.967997593\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.33430223569649, −12.14081415803122, −11.55305128466023, −11.22514418197734, −10.76650236411782, −10.20331670468529, −9.727704871584752, −9.457455803277867, −8.925280395324920, −8.599434362747268, −8.040947863910194, −7.538952051578344, −6.990909853326554, −6.473795244174459, −6.320741277645235, −5.669313636393833, −4.941810610679109, −4.687502245739592, −3.905372811531476, −3.323202029749588, −2.830768540901701, −2.327625444756518, −1.512874435366828, −0.9599893629507264, −0.7561483430953913,
0.7561483430953913, 0.9599893629507264, 1.512874435366828, 2.327625444756518, 2.830768540901701, 3.323202029749588, 3.905372811531476, 4.687502245739592, 4.941810610679109, 5.669313636393833, 6.320741277645235, 6.473795244174459, 6.990909853326554, 7.538952051578344, 8.040947863910194, 8.599434362747268, 8.925280395324920, 9.457455803277867, 9.727704871584752, 10.20331670468529, 10.76650236411782, 11.22514418197734, 11.55305128466023, 12.14081415803122, 12.33430223569649
