| L(s) = 1 | + 2-s + 3.28·3-s + 4-s − 3.78·5-s + 3.28·6-s + 2.49·7-s + 8-s + 7.82·9-s − 3.78·10-s − 11-s + 3.28·12-s − 2.09·13-s + 2.49·14-s − 12.4·15-s + 16-s − 4.28·17-s + 7.82·18-s + 2.91·19-s − 3.78·20-s + 8.19·21-s − 22-s − 1.75·23-s + 3.28·24-s + 9.35·25-s − 2.09·26-s + 15.8·27-s + 2.49·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.89·3-s + 0.5·4-s − 1.69·5-s + 1.34·6-s + 0.941·7-s + 0.353·8-s + 2.60·9-s − 1.19·10-s − 0.301·11-s + 0.949·12-s − 0.580·13-s + 0.665·14-s − 3.21·15-s + 0.250·16-s − 1.03·17-s + 1.84·18-s + 0.667·19-s − 0.847·20-s + 1.78·21-s − 0.213·22-s − 0.364·23-s + 0.671·24-s + 1.87·25-s − 0.410·26-s + 3.05·27-s + 0.470·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.290482905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.290482905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 3.28T + 3T^{2} \) |
| 5 | \( 1 + 3.78T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 - 9.55T + 31T^{2} \) |
| 37 | \( 1 - 9.21T + 37T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 - 6.02T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 9.47T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 7.51T + 79T^{2} \) |
| 83 | \( 1 - 9.30T + 83T^{2} \) |
| 89 | \( 1 + 7.41T + 89T^{2} \) |
| 97 | \( 1 + 3.72T + 97T^{2} \) |
| 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
−7.993648440468779439832429731481, −7.88312136680904011965104518247, −7.31464773299374941169290941372, −6.40929506557057080016678299845, −4.77358230393817940957415792995, −4.45510853720980871897097767676, −3.89924890760225537508966570628, −2.84324758633740715934351019763, −2.52001832052456298226464335277, −1.14273700003129427153976131261,
1.14273700003129427153976131261, 2.52001832052456298226464335277, 2.84324758633740715934351019763, 3.89924890760225537508966570628, 4.45510853720980871897097767676, 4.77358230393817940957415792995, 6.40929506557057080016678299845, 7.31464773299374941169290941372, 7.88312136680904011965104518247, 7.993648440468779439832429731481
