| L(s) = 1 | + 5.15·2-s + 18.5·4-s + 17.0·5-s − 7·7-s + 54.6·8-s + 87.7·10-s − 30.3·11-s − 89.5·13-s − 36.0·14-s + 132.·16-s + 13.4·17-s − 5.37·19-s + 316.·20-s − 156.·22-s + 167.·23-s + 164.·25-s − 461.·26-s − 130.·28-s − 135.·29-s − 18.9·31-s + 248.·32-s + 69.1·34-s − 119.·35-s + 402.·37-s − 27.7·38-s + 929.·40-s − 434.·41-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.32·4-s + 1.52·5-s − 0.377·7-s + 2.41·8-s + 2.77·10-s − 0.831·11-s − 1.91·13-s − 0.689·14-s + 2.07·16-s + 0.191·17-s − 0.0648·19-s + 3.53·20-s − 1.51·22-s + 1.51·23-s + 1.31·25-s − 3.48·26-s − 0.878·28-s − 0.864·29-s − 0.109·31-s + 1.37·32-s + 0.348·34-s − 0.575·35-s + 1.78·37-s − 0.118·38-s + 3.67·40-s − 1.65·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.556707350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.556707350\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.15T + 8T^{2} \) |
| 5 | \( 1 - 17.0T + 125T^{2} \) |
| 11 | \( 1 + 30.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 89.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 13.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.37T + 6.85e3T^{2} \) |
| 23 | \( 1 - 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 18.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 402.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 155.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 412.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 571.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 8.95T + 3.57e5T^{2} \) |
| 73 | \( 1 - 21.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 619.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 259.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 484.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 252.T + 9.12e5T^{2} \) |
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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
−12.73957420952358919817575159077, −11.42030317391092531745809040827, −10.29054800777043781344669274431, −9.484849424574401037762842214895, −7.44585604656292912163016130517, −6.48781478295633333712261820317, −5.41948932847583876857786692293, −4.85238584569912006135070799186, −2.99981241537423370915516635524, −2.15476026271205848448096837071,
2.15476026271205848448096837071, 2.99981241537423370915516635524, 4.85238584569912006135070799186, 5.41948932847583876857786692293, 6.48781478295633333712261820317, 7.44585604656292912163016130517, 9.484849424574401037762842214895, 10.29054800777043781344669274431, 11.42030317391092531745809040827, 12.73957420952358919817575159077
