| L(s) = 1 | − 10.0·3-s − 10.5·7-s + 74.9·9-s + 38.9·11-s + 68.9·13-s − 65.9·17-s − 49.4·19-s + 106.·21-s − 164.·23-s − 484.·27-s − 170.·29-s + 166.·31-s − 392.·33-s + 384.·37-s − 696.·39-s − 22.8·41-s + 136.·43-s − 307.·47-s − 230.·49-s + 665.·51-s − 222·53-s + 499.·57-s + 522.·59-s + 393.·61-s − 793.·63-s − 476.·67-s + 1.66e3·69-s + ⋯ |
| L(s) = 1 | − 1.94·3-s − 0.571·7-s + 2.77·9-s + 1.06·11-s + 1.47·13-s − 0.940·17-s − 0.597·19-s + 1.11·21-s − 1.49·23-s − 3.45·27-s − 1.09·29-s + 0.964·31-s − 2.07·33-s + 1.70·37-s − 2.85·39-s − 0.0868·41-s + 0.484·43-s − 0.954·47-s − 0.672·49-s + 1.82·51-s − 0.575·53-s + 1.16·57-s + 1.15·59-s + 0.824·61-s − 1.58·63-s − 0.869·67-s + 2.89·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8274534308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8274534308\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 10.0T + 27T^{2} \) |
| 7 | \( 1 + 10.5T + 343T^{2} \) |
| 11 | \( 1 - 38.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 65.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 384.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 22.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 136.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 307.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 222T + 1.48e5T^{2} \) |
| 59 | \( 1 - 522.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 476.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 4.26T + 3.57e5T^{2} \) |
| 73 | \( 1 - 601.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 479.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 635.T + 9.12e5T^{2} \) |
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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
−10.02939280312265186343820705539, −9.314845411941267673506053467332, −8.076067684069914391794491977084, −6.73667998768801904671011632693, −6.32400689987465096505527846556, −5.76756571826014662904509485987, −4.42352260392891573715253644440, −3.86259141016086135758025418260, −1.71821238523930940218950753151, −0.58034343836915827891218111286,
0.58034343836915827891218111286, 1.71821238523930940218950753151, 3.86259141016086135758025418260, 4.42352260392891573715253644440, 5.76756571826014662904509485987, 6.32400689987465096505527846556, 6.73667998768801904671011632693, 8.076067684069914391794491977084, 9.314845411941267673506053467332, 10.02939280312265186343820705539
