| L(s) = 1 | + 0.612·5-s − 22.7·7-s − 60.2·11-s − 52.9·13-s − 47.1·17-s + 29.1·19-s − 109.·23-s − 124.·25-s − 10.4·29-s + 220.·31-s − 13.9·35-s − 408.·37-s − 360.·41-s − 236.·43-s + 129.·47-s + 174.·49-s + 117.·53-s − 36.9·55-s + 262.·59-s − 273.·61-s − 32.4·65-s + 89.4·67-s + 350.·71-s − 532.·73-s + 1.37e3·77-s + 166.·79-s + 361.·83-s + ⋯ |
| L(s) = 1 | + 0.0547·5-s − 1.22·7-s − 1.65·11-s − 1.12·13-s − 0.672·17-s + 0.352·19-s − 0.992·23-s − 0.996·25-s − 0.0667·29-s + 1.27·31-s − 0.0672·35-s − 1.81·37-s − 1.37·41-s − 0.838·43-s + 0.400·47-s + 0.508·49-s + 0.305·53-s − 0.0905·55-s + 0.580·59-s − 0.573·61-s − 0.0618·65-s + 0.163·67-s + 0.585·71-s − 0.853·73-s + 2.02·77-s + 0.237·79-s + 0.478·83-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
| \(L(2)\) |
\(\approx\) |
\(0.2115984392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2115984392\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.612T + 125T^{2} \) |
| 7 | \( 1 + 22.7T + 343T^{2} \) |
| 11 | \( 1 + 60.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 117.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 273.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 89.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 532.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 361.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 40.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 614.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.596091450332324717902994772239, −7.86522134237393753697005298420, −7.09939742283050739831832144235, −6.37523218968751705074227228865, −5.45970145332173215031596997178, −4.79996384903510349655687437380, −3.63698197045400441950144076466, −2.78605520462084752714988483343, −2.03086517794332729149153621804, −0.18836308294436167337435587285,
0.18836308294436167337435587285, 2.03086517794332729149153621804, 2.78605520462084752714988483343, 3.63698197045400441950144076466, 4.79996384903510349655687437380, 5.45970145332173215031596997178, 6.37523218968751705074227228865, 7.09939742283050739831832144235, 7.86522134237393753697005298420, 8.596091450332324717902994772239
