L(s) = 1 | − 2.17·2-s − 3.28·4-s − 9.58·5-s + 14.1·7-s + 24.5·8-s + 20.8·10-s − 19.1·11-s + 15.5·13-s − 30.7·14-s − 26.9·16-s − 100.·17-s + 74.3·19-s + 31.4·20-s + 41.5·22-s − 98.9·23-s − 33.2·25-s − 33.7·26-s − 46.5·28-s − 194.·29-s + 52.9·31-s − 137.·32-s + 218.·34-s − 135.·35-s − 212.·37-s − 161.·38-s − 234.·40-s + 395.·41-s + ⋯ |
L(s) = 1 | − 0.767·2-s − 0.410·4-s − 0.856·5-s + 0.764·7-s + 1.08·8-s + 0.657·10-s − 0.524·11-s + 0.331·13-s − 0.586·14-s − 0.420·16-s − 1.43·17-s + 0.897·19-s + 0.351·20-s + 0.402·22-s − 0.896·23-s − 0.265·25-s − 0.254·26-s − 0.313·28-s − 1.24·29-s + 0.306·31-s − 0.760·32-s + 1.10·34-s − 0.655·35-s − 0.945·37-s − 0.689·38-s − 0.928·40-s + 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7274212328\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7274212328\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 - 59T \) |
good | 2 | \( 1 + 2.17T + 8T^{2} \) |
| 5 | \( 1 + 9.58T + 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 11 | \( 1 + 19.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 98.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 52.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 395.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 305.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 630.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 109.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 240.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 100.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 263.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 296.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 626.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 7.08T + 5.71e5T^{2} \) |
| 89 | \( 1 - 132.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−10.49452184855703936413409038854, −9.348833188171388466428978366572, −8.663231880042975895362944983757, −7.79103536238153792922905340822, −7.34299761242873969265351054364, −5.73115767095098912299568948622, −4.59540138514117634833337266535, −3.85042345154651054474684140492, −2.06265471250329851408014089616, −0.58595551638384297048090289261,
0.58595551638384297048090289261, 2.06265471250329851408014089616, 3.85042345154651054474684140492, 4.59540138514117634833337266535, 5.73115767095098912299568948622, 7.34299761242873969265351054364, 7.79103536238153792922905340822, 8.663231880042975895362944983757, 9.348833188171388466428978366572, 10.49452184855703936413409038854