| L(s) = 1 | + 0.236·2-s − 7.94·4-s + 7·7-s − 3.76·8-s + 50.4·11-s + 80.9·13-s + 1.65·14-s + 62.6·16-s + 76.3·17-s + 4.13·19-s + 11.9·22-s − 204.·23-s + 19.1·26-s − 55.6·28-s + 91.1·29-s + 198.·31-s + 44.9·32-s + 18.0·34-s − 155.·37-s + 0.976·38-s + 156.·41-s − 354.·43-s − 400.·44-s − 48.3·46-s − 175.·47-s + 49·49-s − 643.·52-s + ⋯ |
| L(s) = 1 | + 0.0834·2-s − 0.993·4-s + 0.377·7-s − 0.166·8-s + 1.38·11-s + 1.72·13-s + 0.0315·14-s + 0.979·16-s + 1.08·17-s + 0.0499·19-s + 0.115·22-s − 1.85·23-s + 0.144·26-s − 0.375·28-s + 0.583·29-s + 1.14·31-s + 0.248·32-s + 0.0909·34-s − 0.691·37-s + 0.00416·38-s + 0.597·41-s − 1.25·43-s − 1.37·44-s − 0.154·46-s − 0.545·47-s + 0.142·49-s − 1.71·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.340598275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340598275\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 0.236T + 8T^{2} \) |
| 11 | \( 1 - 50.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 80.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.13T + 6.85e3T^{2} \) |
| 23 | \( 1 + 204.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 91.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 354.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 175.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 154.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 734.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 678.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 60.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.28e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 116.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 916.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−8.845604869942392620376622277370, −8.474341455024723470936567120092, −7.68791290025835345030087649929, −6.34629257629547888247689509972, −5.91615389287345527139436811812, −4.79154641553107522513901292180, −3.92538522091243805615370411817, −3.40903490882162041523356912234, −1.61170418332528786560906348292, −0.815818025496236676430602633103,
0.815818025496236676430602633103, 1.61170418332528786560906348292, 3.40903490882162041523356912234, 3.92538522091243805615370411817, 4.79154641553107522513901292180, 5.91615389287345527139436811812, 6.34629257629547888247689509972, 7.68791290025835345030087649929, 8.474341455024723470936567120092, 8.845604869942392620376622277370
