| L(s) = 1 | − 4.76·2-s − 9·3-s − 9.33·4-s − 97.1·5-s + 42.8·6-s − 211.·7-s + 196.·8-s + 81·9-s + 462.·10-s − 633.·11-s + 83.9·12-s − 59.3·13-s + 1.00e3·14-s + 874.·15-s − 638.·16-s − 1.80e3·17-s − 385.·18-s + 1.98e3·19-s + 906.·20-s + 1.90e3·21-s + 3.01e3·22-s + 529·23-s − 1.77e3·24-s + 6.31e3·25-s + 282.·26-s − 729·27-s + 1.97e3·28-s + ⋯ |
| L(s) = 1 | − 0.841·2-s − 0.577·3-s − 0.291·4-s − 1.73·5-s + 0.485·6-s − 1.62·7-s + 1.08·8-s + 0.333·9-s + 1.46·10-s − 1.57·11-s + 0.168·12-s − 0.0974·13-s + 1.37·14-s + 1.00·15-s − 0.623·16-s − 1.51·17-s − 0.280·18-s + 1.26·19-s + 0.506·20-s + 0.940·21-s + 1.32·22-s + 0.208·23-s − 0.627·24-s + 2.01·25-s + 0.0819·26-s − 0.192·27-s + 0.475·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.01599155649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01599155649\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 + 4.76T + 32T^{2} \) |
| 5 | \( 1 + 97.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 211.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 633.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 59.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.80e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.98e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 2.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.38e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.27e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.04e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.13e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.32e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.90e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.42e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.04e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.83e4T + 8.58e9T^{2} \) |
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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
−13.35353201241485333458357999583, −12.64742681033699774361525297663, −11.29793699122157514004126476638, −10.34581345990084781470533299921, −9.156359432938958153404415728324, −7.86177893650285125818737199566, −6.96174750792227027756496718387, −4.92684671229895071140706077797, −3.42889400096023835420422872908, −0.10851163289065741071498041287,
0.10851163289065741071498041287, 3.42889400096023835420422872908, 4.92684671229895071140706077797, 6.96174750792227027756496718387, 7.86177893650285125818737199566, 9.156359432938958153404415728324, 10.34581345990084781470533299921, 11.29793699122157514004126476638, 12.64742681033699774361525297663, 13.35353201241485333458357999583
