| L(s) = 1 | + 7.96·7-s + 44.6·11-s − 68.4·13-s − 90.1·17-s + 75.3·19-s + 11.0·23-s − 145.·29-s + 202.·31-s + 124.·37-s + 150.·41-s + 134.·43-s + 510.·47-s − 279.·49-s + 16.2·53-s + 399.·59-s − 486.·61-s − 353.·67-s − 76.9·71-s + 419.·73-s + 355.·77-s + 110.·79-s − 45.6·83-s + 724.·89-s − 545.·91-s − 539.·97-s + 1.48e3·101-s − 1.20e3·103-s + ⋯ |
| L(s) = 1 | + 0.430·7-s + 1.22·11-s − 1.45·13-s − 1.28·17-s + 0.910·19-s + 0.100·23-s − 0.934·29-s + 1.17·31-s + 0.555·37-s + 0.575·41-s + 0.476·43-s + 1.58·47-s − 0.814·49-s + 0.0422·53-s + 0.880·59-s − 1.02·61-s − 0.644·67-s − 0.128·71-s + 0.673·73-s + 0.526·77-s + 0.157·79-s − 0.0603·83-s + 0.863·89-s − 0.627·91-s − 0.564·97-s + 1.46·101-s − 1.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.185792067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.185792067\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7.96T + 343T^{2} \) |
| 11 | \( 1 - 44.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 11.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 134.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 510.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 16.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 399.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 353.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 76.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 419.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 110.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 45.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 724.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 539.T + 9.12e5T^{2} \) |
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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
−9.178298129036136587915744682817, −8.062354880827993821041988859744, −7.30567230548923089996782741799, −6.63993954781288174983220925881, −5.67668559517736232844178457579, −4.69750764216217429789486008714, −4.10262064777499303399486896794, −2.83556574358795228266912990225, −1.89803907508507037559697429708, −0.69845322659065691321142607095,
0.69845322659065691321142607095, 1.89803907508507037559697429708, 2.83556574358795228266912990225, 4.10262064777499303399486896794, 4.69750764216217429789486008714, 5.67668559517736232844178457579, 6.63993954781288174983220925881, 7.30567230548923089996782741799, 8.062354880827993821041988859744, 9.178298129036136587915744682817
