| L(s) = 1 | − 2.56·3-s + 5-s + 7-s + 3.56·9-s − 2.56·11-s − 0.561·13-s − 2.56·15-s + 4.56·17-s − 2.56·21-s + 25-s − 1.43·27-s − 5.68·29-s − 5.12·31-s + 6.56·33-s + 35-s + 7.12·37-s + 1.43·39-s + 2·41-s − 4·43-s + 3.56·45-s + 11.6·47-s + 49-s − 11.6·51-s + 4.87·53-s − 2.56·55-s + 10.2·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.447·5-s + 0.377·7-s + 1.18·9-s − 0.772·11-s − 0.155·13-s − 0.661·15-s + 1.10·17-s − 0.558·21-s + 0.200·25-s − 0.276·27-s − 1.05·29-s − 0.920·31-s + 1.14·33-s + 0.169·35-s + 1.17·37-s + 0.230·39-s + 0.312·41-s − 0.609·43-s + 0.530·45-s + 1.70·47-s + 0.142·49-s − 1.63·51-s + 0.669·53-s − 0.345·55-s + 1.33·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9937360897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9937360897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 4.87T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−10.05115587833441235708560115896, −9.227358752733090042235253779619, −7.985877485052705884578552726193, −7.26482949375386569656124307747, −6.24950675556098104137628150408, −5.45171691772541142213208612041, −5.09805002191144417980104941382, −3.81025516028958770892901956916, −2.27005271530113938420982767103, −0.820573280747260261696638369161,
0.820573280747260261696638369161, 2.27005271530113938420982767103, 3.81025516028958770892901956916, 5.09805002191144417980104941382, 5.45171691772541142213208612041, 6.24950675556098104137628150408, 7.26482949375386569656124307747, 7.985877485052705884578552726193, 9.227358752733090042235253779619, 10.05115587833441235708560115896
