| L(s) = 1 | + 73.5·2-s + 274.·3-s + 3.35e3·4-s + 7.55e3·5-s + 2.01e4·6-s + 4.67e4·7-s + 9.60e4·8-s − 1.01e5·9-s + 5.55e5·10-s − 9.19e4·11-s + 9.21e5·12-s − 1.62e6·13-s + 3.43e6·14-s + 2.07e6·15-s + 1.92e5·16-s + 5.89e6·17-s − 7.48e6·18-s + 3.33e6·19-s + 2.53e7·20-s + 1.28e7·21-s − 6.76e6·22-s + 2.81e7·23-s + 2.63e7·24-s + 8.22e6·25-s − 1.19e8·26-s − 7.65e7·27-s + 1.56e8·28-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 0.652·3-s + 1.63·4-s + 1.08·5-s + 1.05·6-s + 1.05·7-s + 1.03·8-s − 0.574·9-s + 1.75·10-s − 0.172·11-s + 1.06·12-s − 1.21·13-s + 1.70·14-s + 0.705·15-s + 0.0457·16-s + 1.00·17-s − 0.933·18-s + 0.308·19-s + 1.77·20-s + 0.685·21-s − 0.279·22-s + 0.911·23-s + 0.676·24-s + 0.168·25-s − 1.96·26-s − 1.02·27-s + 1.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(6.910436093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.910436093\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 - 73.5T + 2.04e3T^{2} \) |
| 3 | \( 1 - 274.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 7.55e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.67e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.19e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.62e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 5.89e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 3.33e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.81e7T + 9.52e14T^{2} \) |
| 31 | \( 1 + 1.31e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 8.54e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.41e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.61e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.99e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.74e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.44e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 7.14e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.96e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 8.71e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.16e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.00e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.75e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 6.59e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.15e11T + 7.15e21T^{2} \) |
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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
−14.44367066771072698395491727267, −13.65209692299316775153869470527, −12.44635248240573799447195400550, −11.15925932465366014779357676012, −9.417450719453472076921741260970, −7.61642577581281741774999504756, −5.78474531165647207441727628365, −4.87759754806901132296199477394, −3.04455894450206644634964624506, −1.94878314706909351388013020181,
1.94878314706909351388013020181, 3.04455894450206644634964624506, 4.87759754806901132296199477394, 5.78474531165647207441727628365, 7.61642577581281741774999504756, 9.417450719453472076921741260970, 11.15925932465366014779357676012, 12.44635248240573799447195400550, 13.65209692299316775153869470527, 14.44367066771072698395491727267
