| L(s) = 1 | + 0.470·2-s + 3-s − 1.77·4-s + 4.24·5-s + 0.470·6-s − 3.30·7-s − 1.77·8-s + 9-s + 2·10-s + 1.47·11-s − 1.77·12-s + 0.249·13-s − 1.55·14-s + 4.24·15-s + 2.71·16-s + 5.02·17-s + 0.470·18-s + 2.24·19-s − 7.55·20-s − 3.30·21-s + 0.692·22-s − 6.24·23-s − 1.77·24-s + 13.0·25-s + 0.117·26-s + 27-s + 5.88·28-s + ⋯ |
| L(s) = 1 | + 0.332·2-s + 0.577·3-s − 0.889·4-s + 1.90·5-s + 0.192·6-s − 1.25·7-s − 0.628·8-s + 0.333·9-s + 0.632·10-s + 0.443·11-s − 0.513·12-s + 0.0690·13-s − 0.416·14-s + 1.09·15-s + 0.679·16-s + 1.21·17-s + 0.110·18-s + 0.515·19-s − 1.68·20-s − 0.721·21-s + 0.147·22-s − 1.30·23-s − 0.363·24-s + 2.61·25-s + 0.0229·26-s + 0.192·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.108029681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.108029681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 - 0.249T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 9.71T + 37T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 2.61T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 9.39T + 73T^{2} \) |
| 79 | \( 1 - 0.560T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 4.11T + 89T^{2} \) |
| 97 | \( 1 - 7.19T + 97T^{2} \) |
| 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
−8.456805263820388238056564280176, −7.52845485923915253062538035042, −6.45622858799907861272850292775, −6.04254932732594151305399189556, −5.46158586454560393200599414310, −4.57573334268050956274359079157, −3.51646875049351307350323886523, −3.03673360605539299790139350355, −2.01190645532683727033999057495, −0.925969120298609297117298835419,
0.925969120298609297117298835419, 2.01190645532683727033999057495, 3.03673360605539299790139350355, 3.51646875049351307350323886523, 4.57573334268050956274359079157, 5.46158586454560393200599414310, 6.04254932732594151305399189556, 6.45622858799907861272850292775, 7.52845485923915253062538035042, 8.456805263820388238056564280176
