| L(s) = 1 | + 3.95·2-s + 7.66·4-s + 27.0·7-s − 1.31·8-s − 11·11-s + 82.9·13-s + 106.·14-s − 66.5·16-s + 100.·17-s + 38.6·19-s − 43.5·22-s − 19.5·23-s + 328.·26-s + 207.·28-s + 100.·29-s + 121.·31-s − 252.·32-s + 397.·34-s − 347.·37-s + 152.·38-s − 108.·41-s + 268.·43-s − 84.3·44-s − 77.2·46-s − 568.·47-s + 387.·49-s + 635.·52-s + ⋯ |
| L(s) = 1 | + 1.39·2-s + 0.958·4-s + 1.45·7-s − 0.0581·8-s − 0.301·11-s + 1.76·13-s + 2.04·14-s − 1.03·16-s + 1.43·17-s + 0.466·19-s − 0.421·22-s − 0.176·23-s + 2.47·26-s + 1.39·28-s + 0.641·29-s + 0.706·31-s − 1.39·32-s + 2.00·34-s − 1.54·37-s + 0.652·38-s − 0.412·41-s + 0.950·43-s − 0.288·44-s − 0.247·46-s − 1.76·47-s + 1.13·49-s + 1.69·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.842621135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.842621135\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.95T + 8T^{2} \) |
| 7 | \( 1 - 27.0T + 343T^{2} \) |
| 13 | \( 1 - 82.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 100.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 347.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 268.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 568.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 603.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 32.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 156.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 745.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 334.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 127.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 615.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 579.T + 9.12e5T^{2} \) |
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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
−8.247762888093588158789825329463, −7.975756273919220190283138745468, −6.74990475585717441173734545563, −6.00333939239982826753827290507, −5.22514354733376942798307470958, −4.79230005094268487238453899932, −3.73173077702308127120302444254, −3.20902544333671699606086620562, −1.91082541143573352828673238951, −1.00580124490561771827924890265,
1.00580124490561771827924890265, 1.91082541143573352828673238951, 3.20902544333671699606086620562, 3.73173077702308127120302444254, 4.79230005094268487238453899932, 5.22514354733376942798307470958, 6.00333939239982826753827290507, 6.74990475585717441173734545563, 7.975756273919220190283138745468, 8.247762888093588158789825329463
