| L(s) = 1 | + 2.19·2-s − 3.18·4-s + 5·5-s + 2.76·7-s − 24.5·8-s + 10.9·10-s + 52.6·11-s + 20.4·13-s + 6.06·14-s − 28.3·16-s − 3.66·17-s − 95.6·19-s − 15.9·20-s + 115.·22-s + 89.8·23-s + 25·25-s + 44.8·26-s − 8.80·28-s + 227.·29-s + 279.·31-s + 134.·32-s − 8.03·34-s + 13.8·35-s + 273.·37-s − 209.·38-s − 122.·40-s − 64.8·41-s + ⋯ |
| L(s) = 1 | + 0.775·2-s − 0.398·4-s + 0.447·5-s + 0.149·7-s − 1.08·8-s + 0.346·10-s + 1.44·11-s + 0.436·13-s + 0.115·14-s − 0.443·16-s − 0.0522·17-s − 1.15·19-s − 0.178·20-s + 1.11·22-s + 0.814·23-s + 0.200·25-s + 0.338·26-s − 0.0594·28-s + 1.45·29-s + 1.61·31-s + 0.740·32-s − 0.0405·34-s + 0.0667·35-s + 1.21·37-s − 0.896·38-s − 0.485·40-s − 0.247·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.830012147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.830012147\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 2.19T + 8T^{2} \) |
| 7 | \( 1 - 2.76T + 343T^{2} \) |
| 11 | \( 1 - 52.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 89.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 227.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 273.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 64.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 418.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 197.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 741.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 488.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 310.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 51.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 905.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 663.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 725.T + 9.12e5T^{2} \) |
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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
−10.98680468478441212264617785909, −9.823774917532282220868394819548, −9.009356326545994669929452250141, −8.280438284161399266159875539124, −6.58502333747563300855318437903, −6.12540887557346043583908777185, −4.77479722836887099753179412606, −4.07298826479745293117054001121, −2.77856551342686772519634712802, −1.05306787406970586401934536607,
1.05306787406970586401934536607, 2.77856551342686772519634712802, 4.07298826479745293117054001121, 4.77479722836887099753179412606, 6.12540887557346043583908777185, 6.58502333747563300855318437903, 8.280438284161399266159875539124, 9.009356326545994669929452250141, 9.823774917532282220868394819548, 10.98680468478441212264617785909
