| L(s) = 1 | − 1.21·2-s − 3·3-s − 6.51·4-s + 20.4·5-s + 3.65·6-s − 21.8·7-s + 17.6·8-s + 9·9-s − 24.8·10-s − 6.79·11-s + 19.5·12-s − 83.3·13-s + 26.5·14-s − 61.2·15-s + 30.6·16-s + 55.8·17-s − 10.9·18-s − 133.·20-s + 65.4·21-s + 8.26·22-s − 152.·23-s − 53.0·24-s + 291.·25-s + 101.·26-s − 27·27-s + 142.·28-s + 146.·29-s + ⋯ |
| L(s) = 1 | − 0.430·2-s − 0.577·3-s − 0.814·4-s + 1.82·5-s + 0.248·6-s − 1.17·7-s + 0.780·8-s + 0.333·9-s − 0.785·10-s − 0.186·11-s + 0.470·12-s − 1.77·13-s + 0.506·14-s − 1.05·15-s + 0.478·16-s + 0.797·17-s − 0.143·18-s − 1.48·20-s + 0.679·21-s + 0.0800·22-s − 1.38·23-s − 0.450·24-s + 2.33·25-s + 0.765·26-s − 0.192·27-s + 0.959·28-s + 0.936·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9400195736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9400195736\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 8T^{2} \) |
| 5 | \( 1 - 20.4T + 125T^{2} \) |
| 7 | \( 1 + 21.8T + 343T^{2} \) |
| 11 | \( 1 + 6.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 83.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.8T + 4.91e3T^{2} \) |
| 23 | \( 1 + 152.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 146.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 21.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 364.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 413.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 89.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 100.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 308.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 630.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 568.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.704104268496747038455386770853, −9.090651378821469886740202609520, −7.85116623551494018674035695726, −6.85740437792583000390240209241, −5.99575949867165375085445898329, −5.36097041633453229011291730865, −4.51838439826716508485088498063, −2.99938025421301060791602071004, −1.89285089866204736367277202111, −0.55291287083154897300965693066,
0.55291287083154897300965693066, 1.89285089866204736367277202111, 2.99938025421301060791602071004, 4.51838439826716508485088498063, 5.36097041633453229011291730865, 5.99575949867165375085445898329, 6.85740437792583000390240209241, 7.85116623551494018674035695726, 9.090651378821469886740202609520, 9.704104268496747038455386770853
