| L(s) = 1 | + 3·3-s + 9.34·5-s + 9.07·7-s + 9·9-s + 51.8·11-s + 7.20·13-s + 28.0·15-s + 70.2·17-s − 102.·19-s + 27.2·21-s − 23·23-s − 37.6·25-s + 27·27-s − 11.5·29-s + 189.·31-s + 155.·33-s + 84.7·35-s + 31.2·37-s + 21.6·39-s + 59.0·41-s + 162.·43-s + 84.1·45-s − 82.9·47-s − 260.·49-s + 210.·51-s − 310.·53-s + 484.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.835·5-s + 0.489·7-s + 0.333·9-s + 1.42·11-s + 0.153·13-s + 0.482·15-s + 1.00·17-s − 1.24·19-s + 0.282·21-s − 0.208·23-s − 0.301·25-s + 0.192·27-s − 0.0741·29-s + 1.09·31-s + 0.820·33-s + 0.409·35-s + 0.138·37-s + 0.0887·39-s + 0.224·41-s + 0.576·43-s + 0.278·45-s − 0.257·47-s − 0.760·49-s + 0.578·51-s − 0.804·53-s + 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.391452619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.391452619\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 9.34T + 125T^{2} \) |
| 7 | \( 1 - 9.07T + 343T^{2} \) |
| 11 | \( 1 - 51.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.20T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 189.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 31.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 59.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 162.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 82.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 310.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 368.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 344.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 167.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 667.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 622.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 724.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 114.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
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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
−10.15648704370291215096760778334, −9.511444858026050411255638875108, −8.661271858943684955549838318899, −7.85763988465825810421367633654, −6.63925658976534925271834515488, −5.92078985879072507959697492650, −4.61102393560321749142269468242, −3.60998547083874927380085992995, −2.21200621753170887612442323746, −1.23308400149826877141991572098,
1.23308400149826877141991572098, 2.21200621753170887612442323746, 3.60998547083874927380085992995, 4.61102393560321749142269468242, 5.92078985879072507959697492650, 6.63925658976534925271834515488, 7.85763988465825810421367633654, 8.661271858943684955549838318899, 9.511444858026050411255638875108, 10.15648704370291215096760778334
