| L(s) = 1 | + 2.06·3-s + 28.8·7-s − 22.7·9-s − 18.7·11-s + 86.7·13-s + 64.7·17-s − 27.2·19-s + 59.6·21-s − 102.·23-s − 102.·27-s + 8.87·29-s + 272.·31-s − 38.8·33-s + 82.4·37-s + 179.·39-s − 249.·41-s + 137.·43-s + 439.·47-s + 490.·49-s + 133.·51-s − 490.·53-s − 56.3·57-s + 530.·59-s − 407.·61-s − 655.·63-s + 595.·67-s − 211.·69-s + ⋯ |
| L(s) = 1 | + 0.397·3-s + 1.55·7-s − 0.841·9-s − 0.515·11-s + 1.85·13-s + 0.924·17-s − 0.329·19-s + 0.619·21-s − 0.925·23-s − 0.732·27-s + 0.0568·29-s + 1.58·31-s − 0.204·33-s + 0.366·37-s + 0.735·39-s − 0.948·41-s + 0.486·43-s + 1.36·47-s + 1.42·49-s + 0.367·51-s − 1.27·53-s − 0.130·57-s + 1.17·59-s − 0.855·61-s − 1.31·63-s + 1.08·67-s − 0.368·69-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
| \(L(2)\) |
\(\approx\) |
\(2.925311699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.925311699\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.06T + 27T^{2} \) |
| 7 | \( 1 - 28.8T + 343T^{2} \) |
| 11 | \( 1 + 18.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 64.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.87T + 2.43e4T^{2} \) |
| 31 | \( 1 - 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.4T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 137.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 439.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 490.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 407.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 595.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 569.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 435.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 678.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 711.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.74e3T + 9.12e5T^{2} \) |
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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
−9.931483656050099054603437927879, −8.577495031005727222158176195349, −8.361520482538412468765814851083, −7.64925496537324558470418262640, −6.19080660961479428700203040001, −5.49711489643803989228316229564, −4.41228934065329279660740498706, −3.35988001285977201632444601288, −2.13771642009261616638479911556, −0.998885137406518882225448563297,
0.998885137406518882225448563297, 2.13771642009261616638479911556, 3.35988001285977201632444601288, 4.41228934065329279660740498706, 5.49711489643803989228316229564, 6.19080660961479428700203040001, 7.64925496537324558470418262640, 8.361520482538412468765814851083, 8.577495031005727222158176195349, 9.931483656050099054603437927879
