| L(s) = 1 | + 2-s + 2.04·3-s + 4-s + 3.63·5-s + 2.04·6-s − 0.447·7-s + 8-s + 1.19·9-s + 3.63·10-s + 11-s + 2.04·12-s − 1.53·13-s − 0.447·14-s + 7.45·15-s + 16-s − 0.931·17-s + 1.19·18-s + 4.19·19-s + 3.63·20-s − 0.917·21-s + 22-s + 3.62·23-s + 2.04·24-s + 8.22·25-s − 1.53·26-s − 3.68·27-s − 0.447·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.18·3-s + 0.5·4-s + 1.62·5-s + 0.836·6-s − 0.169·7-s + 0.353·8-s + 0.399·9-s + 1.15·10-s + 0.301·11-s + 0.591·12-s − 0.424·13-s − 0.119·14-s + 1.92·15-s + 0.250·16-s − 0.225·17-s + 0.282·18-s + 0.961·19-s + 0.813·20-s − 0.200·21-s + 0.213·22-s + 0.756·23-s + 0.418·24-s + 1.64·25-s − 0.300·26-s − 0.710·27-s − 0.0845·28-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
| \(L(1)\) |
\(\approx\) |
\(6.344439054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.344439054\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 + 0.447T + 7T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 + 0.931T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 + 8.70T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 + 3.32T + 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 - 4.07T + 59T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 - 5.06T + 67T^{2} \) |
| 71 | \( 1 - 7.80T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 0.959T + 97T^{2} \) |
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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
−8.580359572835999737424511586183, −7.46540159934716894691206686721, −6.94242596916741444279959306882, −6.03505448485555123918854502593, −5.45197312834452616559670987226, −4.68782308913732858514154606885, −3.56042992656126232322720879700, −2.87665985493927703854087937905, −2.22561653800590874930730993222, −1.39052895096283966412056270135,
1.39052895096283966412056270135, 2.22561653800590874930730993222, 2.87665985493927703854087937905, 3.56042992656126232322720879700, 4.68782308913732858514154606885, 5.45197312834452616559670987226, 6.03505448485555123918854502593, 6.94242596916741444279959306882, 7.46540159934716894691206686721, 8.580359572835999737424511586183
