| L(s) = 1 | − 1.56·2-s + 3·3-s − 5.56·4-s − 5·5-s − 4.68·6-s + 10.2·7-s + 21.1·8-s + 9·9-s + 7.80·10-s − 16.6·12-s + 40.8·13-s − 16·14-s − 15·15-s + 11.4·16-s + 98.7·17-s − 14.0·18-s + 39.6·19-s + 27.8·20-s + 30.7·21-s + 61.6·23-s + 63.5·24-s + 25·25-s − 63.8·26-s + 27·27-s − 56.9·28-s + 149.·29-s + 23.4·30-s + ⋯ |
| L(s) = 1 | − 0.552·2-s + 0.577·3-s − 0.695·4-s − 0.447·5-s − 0.318·6-s + 0.553·7-s + 0.935·8-s + 0.333·9-s + 0.246·10-s − 0.401·12-s + 0.872·13-s − 0.305·14-s − 0.258·15-s + 0.178·16-s + 1.40·17-s − 0.184·18-s + 0.478·19-s + 0.310·20-s + 0.319·21-s + 0.559·23-s + 0.540·24-s + 0.200·25-s − 0.481·26-s + 0.192·27-s − 0.384·28-s + 0.954·29-s + 0.142·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.003615445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.003615445\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 8T^{2} \) |
| 7 | \( 1 - 10.2T + 343T^{2} \) |
| 13 | \( 1 - 40.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 98.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 39.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 44.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.36T + 7.95e4T^{2} \) |
| 47 | \( 1 + 333.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 640.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 370.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 714.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 404.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 939.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 951.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 735.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 385.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 966.T + 9.12e5T^{2} \) |
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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
−8.626895943751117484650205134830, −8.329416802262843097606747714239, −7.66648913910745085004576431271, −6.80865577436549044515143411398, −5.50761729835406809003913038289, −4.75949872444302853093417180656, −3.82699095849970679249022270877, −3.07576153527720861045827526549, −1.52701016907896674242472741761, −0.790390644580710781391790215407,
0.790390644580710781391790215407, 1.52701016907896674242472741761, 3.07576153527720861045827526549, 3.82699095849970679249022270877, 4.75949872444302853093417180656, 5.50761729835406809003913038289, 6.80865577436549044515143411398, 7.66648913910745085004576431271, 8.329416802262843097606747714239, 8.626895943751117484650205134830
