| L(s) = 1 | − 6.97·3-s + 5·5-s + 7·7-s + 21.6·9-s + 28.7·11-s + 34.9·13-s − 34.8·15-s − 80.1·17-s + 86.6·19-s − 48.8·21-s + 208.·23-s + 25·25-s + 37.2·27-s − 207.·29-s − 44.6·31-s − 200.·33-s + 35·35-s + 98.0·37-s − 244.·39-s + 153.·41-s − 269.·43-s + 108.·45-s + 176.·47-s + 49·49-s + 559.·51-s − 518.·53-s + 143.·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 0.447·5-s + 0.377·7-s + 0.802·9-s + 0.786·11-s + 0.746·13-s − 0.600·15-s − 1.14·17-s + 1.04·19-s − 0.507·21-s + 1.89·23-s + 0.200·25-s + 0.265·27-s − 1.32·29-s − 0.258·31-s − 1.05·33-s + 0.169·35-s + 0.435·37-s − 1.00·39-s + 0.584·41-s − 0.956·43-s + 0.358·45-s + 0.547·47-s + 0.142·49-s + 1.53·51-s − 1.34·53-s + 0.351·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.561693316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561693316\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 + 6.97T + 27T^{2} \) |
| 11 | \( 1 - 28.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 208.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 44.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 153.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 269.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 176.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 518.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 226.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 696.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 653.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 731.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 697.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 577.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.00e3T + 9.12e5T^{2} \) |
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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
−9.338074852956621695920623592581, −8.936256872254914607987939173182, −7.58657775339464985052542362648, −6.69911591441218561159661241017, −6.08376302552874378039297408291, −5.23710945209461408666642715358, −4.53101326602752208994818283271, −3.26123447640942991623764409722, −1.68650183081710239312541799792, −0.73759663048544458657972328074,
0.73759663048544458657972328074, 1.68650183081710239312541799792, 3.26123447640942991623764409722, 4.53101326602752208994818283271, 5.23710945209461408666642715358, 6.08376302552874378039297408291, 6.69911591441218561159661241017, 7.58657775339464985052542362648, 8.936256872254914607987939173182, 9.338074852956621695920623592581
