| L(s) = 1 | − 2.49·2-s + 20.1·3-s − 25.7·4-s − 25·5-s − 50.1·6-s − 99.5·7-s + 144.·8-s + 161.·9-s + 62.3·10-s + 121·11-s − 518.·12-s + 508.·13-s + 248.·14-s − 503.·15-s + 466.·16-s − 717.·17-s − 403.·18-s − 361·19-s + 644.·20-s − 2.00e3·21-s − 301.·22-s + 3.74e3·23-s + 2.89e3·24-s + 625·25-s − 1.26e3·26-s − 1.63e3·27-s + 2.56e3·28-s + ⋯ |
| L(s) = 1 | − 0.440·2-s + 1.29·3-s − 0.805·4-s − 0.447·5-s − 0.568·6-s − 0.767·7-s + 0.795·8-s + 0.666·9-s + 0.197·10-s + 0.301·11-s − 1.04·12-s + 0.835·13-s + 0.338·14-s − 0.577·15-s + 0.455·16-s − 0.602·17-s − 0.293·18-s − 0.229·19-s + 0.360·20-s − 0.991·21-s − 0.132·22-s + 1.47·23-s + 1.02·24-s + 0.200·25-s − 0.367·26-s − 0.430·27-s + 0.618·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.603081101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.603081101\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 2.49T + 32T^{2} \) |
| 3 | \( 1 - 20.1T + 243T^{2} \) |
| 7 | \( 1 + 99.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 508.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 717.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.92e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.21e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.75e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 9.37e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.92e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.09e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.62e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.08e5T + 8.58e9T^{2} \) |
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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
−9.020344206847372600494940110525, −8.675555512678567064541843529929, −7.73556985063903480898195747565, −7.03147683005622975238349780116, −5.82734057201586064839873985790, −4.54173734062406169523388276898, −3.69837924686841520806071540998, −3.09138675041749798459351092758, −1.76079089877726121792885559569, −0.55490500781532784825838589593,
0.55490500781532784825838589593, 1.76079089877726121792885559569, 3.09138675041749798459351092758, 3.69837924686841520806071540998, 4.54173734062406169523388276898, 5.82734057201586064839873985790, 7.03147683005622975238349780116, 7.73556985063903480898195747565, 8.675555512678567064541843529929, 9.020344206847372600494940110525
