| L(s) = 1 | − 18.5·5-s + 29.7·7-s − 9.16·11-s + 80.3·13-s + 31.1·17-s − 89.8·19-s + 57.1·23-s + 220.·25-s − 167.·29-s − 270.·31-s − 552.·35-s − 157.·37-s + 404.·41-s + 317.·43-s − 63.1·47-s + 542.·49-s − 616.·53-s + 170.·55-s + 137.·59-s + 200.·61-s − 1.49e3·65-s + 576.·67-s + 305.·71-s − 198.·73-s − 272.·77-s − 335.·79-s + 981.·83-s + ⋯ |
| L(s) = 1 | − 1.66·5-s + 1.60·7-s − 0.251·11-s + 1.71·13-s + 0.444·17-s − 1.08·19-s + 0.518·23-s + 1.76·25-s − 1.06·29-s − 1.56·31-s − 2.66·35-s − 0.699·37-s + 1.53·41-s + 1.12·43-s − 0.196·47-s + 1.58·49-s − 1.59·53-s + 0.417·55-s + 0.304·59-s + 0.420·61-s − 2.84·65-s + 1.05·67-s + 0.510·71-s − 0.319·73-s − 0.403·77-s − 0.477·79-s + 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.823621766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.823621766\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 18.5T + 125T^{2} \) |
| 7 | \( 1 - 29.7T + 343T^{2} \) |
| 11 | \( 1 + 9.16T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 57.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 157.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 404.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 317.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 63.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 616.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 200.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 576.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 305.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 198.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 335.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 981.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 51.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 678.T + 9.12e5T^{2} \) |
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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
−9.051076537154053461325717785022, −8.463421864508667300353302235310, −7.82438431401185570579979920600, −7.28743052153791208339737046560, −5.98851157908469517992082894687, −4.97054272422181931386519103495, −4.09667582164504291110653221502, −3.50093336545168443961977673665, −1.87073052382176919166684770099, −0.72281935437579932846874203024,
0.72281935437579932846874203024, 1.87073052382176919166684770099, 3.50093336545168443961977673665, 4.09667582164504291110653221502, 4.97054272422181931386519103495, 5.98851157908469517992082894687, 7.28743052153791208339737046560, 7.82438431401185570579979920600, 8.463421864508667300353302235310, 9.051076537154053461325717785022
