| L(s) = 1 | + 7.21·3-s + 20.3·5-s + 22.2·7-s + 25.0·9-s + 10.3·11-s − 62.7·13-s + 146.·15-s + 70.8·17-s + 13.5·19-s + 160.·21-s − 168.·23-s + 288.·25-s − 13.7·27-s − 29·29-s + 116.·31-s + 74.5·33-s + 452.·35-s − 270.·37-s − 453.·39-s + 307.·41-s + 377.·43-s + 510.·45-s + 65.4·47-s + 151.·49-s + 511.·51-s + 28.0·53-s + 210.·55-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 1.81·5-s + 1.20·7-s + 0.929·9-s + 0.283·11-s − 1.33·13-s + 2.52·15-s + 1.01·17-s + 0.163·19-s + 1.66·21-s − 1.52·23-s + 2.30·25-s − 0.0979·27-s − 0.185·29-s + 0.675·31-s + 0.393·33-s + 2.18·35-s − 1.20·37-s − 1.86·39-s + 1.17·41-s + 1.34·43-s + 1.69·45-s + 0.203·47-s + 0.442·49-s + 1.40·51-s + 0.0725·53-s + 0.515·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.455608180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.455608180\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 7.21T + 27T^{2} \) |
| 5 | \( 1 - 20.3T + 125T^{2} \) |
| 7 | \( 1 - 22.2T + 343T^{2} \) |
| 11 | \( 1 - 10.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 70.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 168.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 116.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 377.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 65.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 28.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 766.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 565.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 220.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 815.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 716.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 785.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−9.056319177768136558270266459452, −8.021492379049230625043601991645, −7.66602333680935821115368223587, −6.51074674568038235100061597212, −5.54270601794037894097754272690, −4.91669926268518353546145448110, −3.76018969838078524718042861339, −2.47481691454535239245704202039, −2.16429600864814904700638763553, −1.23086505636409963049789278499,
1.23086505636409963049789278499, 2.16429600864814904700638763553, 2.47481691454535239245704202039, 3.76018969838078524718042861339, 4.91669926268518353546145448110, 5.54270601794037894097754272690, 6.51074674568038235100061597212, 7.66602333680935821115368223587, 8.021492379049230625043601991645, 9.056319177768136558270266459452
