| L(s) = 1 | + 2·2-s + 4·4-s − 13.9·5-s + 2.93·7-s + 8·8-s − 27.8·10-s + 23.9·13-s + 5.87·14-s + 16·16-s + 33.6·17-s − 156.·19-s − 55.7·20-s + 77.6·23-s + 69.3·25-s + 47.8·26-s + 11.7·28-s + 160.·29-s + 182.·31-s + 32·32-s + 67.3·34-s − 40.9·35-s − 272.·37-s − 313.·38-s − 111.·40-s + 211.·41-s + 251.·43-s + 155.·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.24·5-s + 0.158·7-s + 0.353·8-s − 0.881·10-s + 0.510·13-s + 0.112·14-s + 0.250·16-s + 0.480·17-s − 1.89·19-s − 0.623·20-s + 0.703·23-s + 0.554·25-s + 0.361·26-s + 0.0793·28-s + 1.02·29-s + 1.05·31-s + 0.176·32-s + 0.339·34-s − 0.197·35-s − 1.21·37-s − 1.33·38-s − 0.440·40-s + 0.805·41-s + 0.890·43-s + 0.497·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 13.9T + 125T^{2} \) |
| 7 | \( 1 - 2.93T + 343T^{2} \) |
| 13 | \( 1 - 23.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 33.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 156.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 182.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 272.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 211.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 251.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 477.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 557.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 436.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 310.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 367.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 787.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 727.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 463.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 447.T + 9.12e5T^{2} \) |
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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
−8.292755185957156239992742574580, −7.52383863551271078764289959155, −6.68686449874922510852955082227, −6.01867460221539230284337972415, −4.84280077002203727829995829514, −4.28889115177298555160371916383, −3.50832281339086443509907146523, −2.61011749440877757683366185608, −1.28381347642813954922418770951, 0,
1.28381347642813954922418770951, 2.61011749440877757683366185608, 3.50832281339086443509907146523, 4.28889115177298555160371916383, 4.84280077002203727829995829514, 6.01867460221539230284337972415, 6.68686449874922510852955082227, 7.52383863551271078764289959155, 8.292755185957156239992742574580
