| L(s) = 1 | + 2.30·2-s − 2.69·4-s + 5·5-s + 33.7·7-s − 24.6·8-s + 11.5·10-s + 11·11-s + 37.9·13-s + 77.6·14-s − 35.1·16-s − 44.2·17-s − 42.3·19-s − 13.4·20-s + 25.3·22-s + 30.6·23-s + 25·25-s + 87.2·26-s − 90.9·28-s + 194.·29-s − 82.9·31-s + 116.·32-s − 101.·34-s + 168.·35-s + 445.·37-s − 97.6·38-s − 123.·40-s + 206.·41-s + ⋯ |
| L(s) = 1 | + 0.814·2-s − 0.337·4-s + 0.447·5-s + 1.82·7-s − 1.08·8-s + 0.364·10-s + 0.301·11-s + 0.808·13-s + 1.48·14-s − 0.548·16-s − 0.631·17-s − 0.511·19-s − 0.150·20-s + 0.245·22-s + 0.278·23-s + 0.200·25-s + 0.658·26-s − 0.614·28-s + 1.24·29-s − 0.480·31-s + 0.641·32-s − 0.514·34-s + 0.814·35-s + 1.97·37-s − 0.416·38-s − 0.486·40-s + 0.785·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.446204822\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.446204822\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 2.30T + 8T^{2} \) |
| 7 | \( 1 - 33.7T + 343T^{2} \) |
| 13 | \( 1 - 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 82.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 445.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 382.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 771.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 210.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 452.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 310.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 174.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.19e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 613.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.20e3T + 9.12e5T^{2} \) |
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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
−10.89249386386112982429566457474, −9.536896115982664210555517395144, −8.646912713165241223533809141844, −8.020525354473909591309746658994, −6.55986353452261464002826945661, −5.63734274543036230687894354025, −4.71082433696483487810472430694, −4.07765733899993420692032777261, −2.49467747302187274090527518882, −1.13331571675258912952629822089,
1.13331571675258912952629822089, 2.49467747302187274090527518882, 4.07765733899993420692032777261, 4.71082433696483487810472430694, 5.63734274543036230687894354025, 6.55986353452261464002826945661, 8.020525354473909591309746658994, 8.646912713165241223533809141844, 9.536896115982664210555517395144, 10.89249386386112982429566457474
