| L(s) = 1 | + 1.73·2-s − 0.732·3-s + 0.999·4-s − 5-s − 1.26·6-s + 2·7-s − 1.73·8-s − 2.46·9-s − 1.73·10-s − 1.26·11-s − 0.732·12-s + 13-s + 3.46·14-s + 0.732·15-s − 5·16-s + 3.46·17-s − 4.26·18-s + 4.19·19-s − 0.999·20-s − 1.46·21-s − 2.19·22-s + 4.73·23-s + 1.26·24-s + 25-s + 1.73·26-s + 4·27-s + 1.99·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.422·3-s + 0.499·4-s − 0.447·5-s − 0.517·6-s + 0.755·7-s − 0.612·8-s − 0.821·9-s − 0.547·10-s − 0.382·11-s − 0.211·12-s + 0.277·13-s + 0.925·14-s + 0.189·15-s − 1.25·16-s + 0.840·17-s − 1.00·18-s + 0.962·19-s − 0.223·20-s − 0.319·21-s − 0.468·22-s + 0.986·23-s + 0.258·24-s + 0.200·25-s + 0.339·26-s + 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245764268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245764268\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.26T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 0.196T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 15.1T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−14.66513874748013962570787213521, −13.92119611272951525106243315187, −12.69940336201952227391431771443, −11.71547383331981146360180340762, −10.97194024744893244170118313479, −9.036330438579938888990771144807, −7.58651044449099949588254922570, −5.80996823928595783242142273545, −4.93498568789334957600638770142, −3.30278022896545756679526281316,
3.30278022896545756679526281316, 4.93498568789334957600638770142, 5.80996823928595783242142273545, 7.58651044449099949588254922570, 9.036330438579938888990771144807, 10.97194024744893244170118313479, 11.71547383331981146360180340762, 12.69940336201952227391431771443, 13.92119611272951525106243315187, 14.66513874748013962570787213521
