| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 11-s + 12-s − 13-s + 15-s + 16-s + 3·17-s + 18-s − 6·19-s + 20-s − 22-s + 23-s + 24-s + 25-s − 26-s + 27-s + 6·29-s + 30-s − 8·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.000004154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.000004154\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.67198803906613, −14.91941609600402, −14.63356760652523, −14.12333608792631, −13.58676065939378, −12.98572679609509, −12.56813383735967, −12.13000028285676, −11.28870204053370, −10.68229061850528, −10.20987510036941, −9.677150313429094, −8.806974756466253, −8.491830487608978, −7.649562605076903, −7.110648207878823, −6.523957589960138, −5.762855133498003, −5.259695784251130, −4.517406425741243, −3.858576388039586, −3.192998608439874, −2.372720094070882, −1.946043771163756, −0.7939432884723676,
0.7939432884723676, 1.946043771163756, 2.372720094070882, 3.192998608439874, 3.858576388039586, 4.517406425741243, 5.259695784251130, 5.762855133498003, 6.523957589960138, 7.110648207878823, 7.649562605076903, 8.491830487608978, 8.806974756466253, 9.677150313429094, 10.20987510036941, 10.68229061850528, 11.28870204053370, 12.13000028285676, 12.56813383735967, 12.98572679609509, 13.58676065939378, 14.12333608792631, 14.63356760652523, 14.91941609600402, 15.67198803906613
