| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 8.35e3·5-s − 1.68e4·7-s + 3.27e4·8-s − 2.67e5·10-s + 1.55e5·11-s + 2.49e6·13-s − 5.37e5·14-s + 1.04e6·16-s − 6.27e6·17-s − 1.54e7·19-s − 8.55e6·20-s + 4.98e6·22-s − 3.08e7·23-s + 2.09e7·25-s + 7.99e7·26-s − 1.72e7·28-s − 5.84e7·29-s + 7.55e7·31-s + 3.35e7·32-s − 2.00e8·34-s + 1.40e8·35-s − 1.85e8·37-s − 4.95e8·38-s − 2.73e8·40-s + 9.65e8·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.19·5-s − 0.377·7-s + 0.353·8-s − 0.845·10-s + 0.291·11-s + 1.86·13-s − 0.267·14-s + 0.250·16-s − 1.07·17-s − 1.43·19-s − 0.597·20-s + 0.206·22-s − 1.00·23-s + 0.428·25-s + 1.31·26-s − 0.188·28-s − 0.529·29-s + 0.474·31-s + 0.176·32-s − 0.757·34-s + 0.451·35-s − 0.440·37-s − 1.01·38-s − 0.422·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.410334193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.410334193\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 + 8.35e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 1.55e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.49e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.27e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.54e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.08e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 5.84e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.55e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 1.85e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.65e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.23e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.09e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.38e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 9.92e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 5.12e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.09e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.18e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 3.38e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.03e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 9.58e8T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.95e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 6.10e10T + 7.15e21T^{2} \) |
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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
−11.28613500970475313468387485608, −10.68365007755501456319387826110, −8.932105679294117239251410088297, −8.046773103372518180511283245277, −6.75528910672348954624539045431, −5.90427359552701146887256509763, −4.07350545654179281405584941490, −3.89481475110773531852345349148, −2.26540978030759939111915321010, −0.66905170900574004297724662704,
0.66905170900574004297724662704, 2.26540978030759939111915321010, 3.89481475110773531852345349148, 4.07350545654179281405584941490, 5.90427359552701146887256509763, 6.75528910672348954624539045431, 8.046773103372518180511283245277, 8.932105679294117239251410088297, 10.68365007755501456319387826110, 11.28613500970475313468387485608
