| L(s) = 1 | + 9.15·5-s + 27.4·7-s − 20.5·11-s + 32.0·13-s + 111.·17-s + 129.·19-s − 9.16·23-s − 41.1·25-s − 41.0·29-s + 187.·31-s + 251.·35-s − 114.·37-s + 282.·41-s − 89.3·43-s − 54.6·47-s + 408.·49-s − 726.·53-s − 187.·55-s + 216.·59-s + 754.·61-s + 293.·65-s + 379.·67-s − 302.·71-s + 504.·73-s − 562.·77-s − 301.·79-s − 599.·83-s + ⋯ |
| L(s) = 1 | + 0.818·5-s + 1.48·7-s − 0.562·11-s + 0.683·13-s + 1.59·17-s + 1.56·19-s − 0.0830·23-s − 0.329·25-s − 0.262·29-s + 1.08·31-s + 1.21·35-s − 0.507·37-s + 1.07·41-s − 0.317·43-s − 0.169·47-s + 1.19·49-s − 1.88·53-s − 0.460·55-s + 0.477·59-s + 1.58·61-s + 0.559·65-s + 0.691·67-s − 0.504·71-s + 0.808·73-s − 0.832·77-s − 0.429·79-s − 0.792·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.960193942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.960193942\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 9.15T + 125T^{2} \) |
| 7 | \( 1 - 27.4T + 343T^{2} \) |
| 11 | \( 1 + 20.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 111.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9.16T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 187.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 89.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 54.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 726.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 754.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 504.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 301.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 599.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 277.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 765.T + 9.12e5T^{2} \) |
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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
−8.460827656502033814586596566504, −7.945066488648513771639825870649, −7.31804025920619724697011036400, −6.09791993292967328864996146751, −5.41765775799692119809725133327, −4.95143584382472748902946692083, −3.74580905898083027413316148088, −2.73650201248061846516988857273, −1.63165942303202124481769589764, −0.997978014035165967829543820555,
0.997978014035165967829543820555, 1.63165942303202124481769589764, 2.73650201248061846516988857273, 3.74580905898083027413316148088, 4.95143584382472748902946692083, 5.41765775799692119809725133327, 6.09791993292967328864996146751, 7.31804025920619724697011036400, 7.945066488648513771639825870649, 8.460827656502033814586596566504
