| L(s) = 1 | − 4.42·2-s + 11.5·4-s + 7·7-s − 15.7·8-s + 23.3·11-s + 3.56·13-s − 30.9·14-s − 22.7·16-s + 57.5·17-s − 51.1·19-s − 103.·22-s + 65.6·23-s − 15.7·26-s + 80.9·28-s − 41.6·29-s − 167.·31-s + 226.·32-s − 254.·34-s − 224.·37-s + 226.·38-s − 196.·41-s + 58.9·43-s + 270.·44-s − 290.·46-s + 41.9·47-s + 49·49-s + 41.2·52-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.44·4-s + 0.377·7-s − 0.697·8-s + 0.640·11-s + 0.0761·13-s − 0.591·14-s − 0.354·16-s + 0.820·17-s − 0.617·19-s − 1.00·22-s + 0.595·23-s − 0.119·26-s + 0.546·28-s − 0.266·29-s − 0.970·31-s + 1.25·32-s − 1.28·34-s − 0.997·37-s + 0.965·38-s − 0.749·41-s + 0.209·43-s + 0.926·44-s − 0.930·46-s + 0.130·47-s + 0.142·49-s + 0.110·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 4.42T + 8T^{2} \) |
| 11 | \( 1 - 23.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.56T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 65.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 41.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 224.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 58.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 41.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 33.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 700.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 453.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 370.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 54.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 430.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 737.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 150.T + 9.12e5T^{2} \) |
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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
−8.856313701386018720959969105333, −7.990673751888123906285450083875, −7.31761093192457734274122590742, −6.59271941974846036949428184158, −5.55685221795902748955977923649, −4.43962317105888616865172870694, −3.27150044087074957163924660323, −1.95332170470552353019557658833, −1.19764643516750001531035071499, 0,
1.19764643516750001531035071499, 1.95332170470552353019557658833, 3.27150044087074957163924660323, 4.43962317105888616865172870694, 5.55685221795902748955977923649, 6.59271941974846036949428184158, 7.31761093192457734274122590742, 7.990673751888123906285450083875, 8.856313701386018720959969105333
