| L(s) = 1 | − 1.28·2-s − 6.36·4-s − 16.8·5-s + 7·7-s + 18.3·8-s + 21.6·10-s − 11·11-s − 68.0·13-s − 8.96·14-s + 27.3·16-s − 119.·17-s + 16.8·19-s + 107.·20-s + 14.0·22-s − 199.·23-s + 160.·25-s + 87.1·26-s − 44.5·28-s − 181.·29-s − 31.5·31-s − 182.·32-s + 152.·34-s − 118.·35-s + 75.9·37-s − 21.5·38-s − 310.·40-s − 408.·41-s + ⋯ |
| L(s) = 1 | − 0.452·2-s − 0.795·4-s − 1.51·5-s + 0.377·7-s + 0.812·8-s + 0.683·10-s − 0.301·11-s − 1.45·13-s − 0.171·14-s + 0.427·16-s − 1.69·17-s + 0.203·19-s + 1.20·20-s + 0.136·22-s − 1.81·23-s + 1.28·25-s + 0.657·26-s − 0.300·28-s − 1.15·29-s − 0.182·31-s − 1.00·32-s + 0.768·34-s − 0.570·35-s + 0.337·37-s − 0.0920·38-s − 1.22·40-s − 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1542527293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1542527293\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.28T + 8T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 13 | \( 1 + 68.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 199.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 75.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 408.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 97.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 563.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 224.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 622.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 280.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 807.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 710.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 191.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 816.T + 9.12e5T^{2} \) |
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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
−9.991514417338405308009689847012, −9.112631660369069391587491690243, −8.200915202396081111538123425956, −7.74693263999960597908787248873, −6.89538536396294361612638216849, −5.25023589704711290283776898161, −4.44548886330974979044271168391, −3.75865893652698915563944941657, −2.11601982179219386698725466801, −0.23267697553066684775333215359,
0.23267697553066684775333215359, 2.11601982179219386698725466801, 3.75865893652698915563944941657, 4.44548886330974979044271168391, 5.25023589704711290283776898161, 6.89538536396294361612638216849, 7.74693263999960597908787248873, 8.200915202396081111538123425956, 9.112631660369069391587491690243, 9.991514417338405308009689847012
