| L(s) = 1 | − 2.56·3-s − 0.561·5-s + 3.56·9-s − 11-s − 3.12·13-s + 1.43·15-s + 2·17-s − 5.12·19-s + 1.43·23-s − 4.68·25-s − 1.43·27-s − 2·29-s − 10.5·31-s + 2.56·33-s + 4.56·37-s + 8·39-s + 10·41-s − 4·43-s − 2.00·45-s − 6.24·47-s − 5.12·51-s − 4.24·53-s + 0.561·55-s + 13.1·57-s − 5.43·59-s + 4.24·61-s + 1.75·65-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.251·5-s + 1.18·9-s − 0.301·11-s − 0.866·13-s + 0.371·15-s + 0.485·17-s − 1.17·19-s + 0.299·23-s − 0.936·25-s − 0.276·27-s − 0.371·29-s − 1.89·31-s + 0.445·33-s + 0.749·37-s + 1.28·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 0.911·47-s − 0.717·51-s − 0.583·53-s + 0.0757·55-s + 1.73·57-s − 0.708·59-s + 0.543·61-s + 0.217·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3629116473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3629116473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 - 2.56T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.56T + 97T^{2} \) |
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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
−7.58938178859091446100141818697, −7.05304993261389219791783696386, −6.24681985257342302451242360293, −5.72126365569403913800766652732, −5.09634566216679401783634493761, −4.44085152021846122170457395216, −3.68963601465300729979918360750, −2.55162866280063207870795350546, −1.57918839345820614247782915445, −0.31640324540219889555662500084,
0.31640324540219889555662500084, 1.57918839345820614247782915445, 2.55162866280063207870795350546, 3.68963601465300729979918360750, 4.44085152021846122170457395216, 5.09634566216679401783634493761, 5.72126365569403913800766652732, 6.24681985257342302451242360293, 7.05304993261389219791783696386, 7.58938178859091446100141818697
