| L(s) = 1 | − 2-s + 4-s + 2.82·5-s + 4.56·7-s − 8-s − 2.82·10-s − 3.18·11-s + 1.77·13-s − 4.56·14-s + 16-s + 4.40·17-s + 0.0741·19-s + 2.82·20-s + 3.18·22-s + 1.08·23-s + 2.96·25-s − 1.77·26-s + 4.56·28-s − 0.601·29-s + 1.68·31-s − 32-s − 4.40·34-s + 12.8·35-s − 4.33·37-s − 0.0741·38-s − 2.82·40-s − 9.64·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.26·5-s + 1.72·7-s − 0.353·8-s − 0.892·10-s − 0.961·11-s + 0.491·13-s − 1.22·14-s + 0.250·16-s + 1.06·17-s + 0.0169·19-s + 0.631·20-s + 0.680·22-s + 0.227·23-s + 0.593·25-s − 0.347·26-s + 0.863·28-s − 0.111·29-s + 0.301·31-s − 0.176·32-s − 0.755·34-s + 2.18·35-s − 0.712·37-s − 0.0120·38-s − 0.446·40-s − 1.50·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.367862027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.367862027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 269 | \( 1 + T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 17 | \( 1 - 4.40T + 17T^{2} \) |
| 19 | \( 1 - 0.0741T + 19T^{2} \) |
| 23 | \( 1 - 1.08T + 23T^{2} \) |
| 29 | \( 1 + 0.601T + 29T^{2} \) |
| 31 | \( 1 - 1.68T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 - 3.25T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 - 4.39T + 53T^{2} \) |
| 59 | \( 1 - 3.57T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 2.38T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 - 6.40T + 97T^{2} \) |
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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
−8.415764203858847382314777751937, −7.70196492072986438318824043104, −7.00854608408775571590688006181, −6.00917134663444916037171979053, −5.33363253552634405264011547835, −4.95730997329266574617656117772, −3.61429808159170046385790302333, −2.45096753613703823240786191438, −1.81743832117265610324458096705, −1.01674262567008595707802289384,
1.01674262567008595707802289384, 1.81743832117265610324458096705, 2.45096753613703823240786191438, 3.61429808159170046385790302333, 4.95730997329266574617656117772, 5.33363253552634405264011547835, 6.00917134663444916037171979053, 7.00854608408775571590688006181, 7.70196492072986438318824043104, 8.415764203858847382314777751937
