| L(s) = 1 | + 3.61·2-s + 5.03·4-s + 11.1·5-s − 7·7-s − 10.6·8-s + 40.2·10-s + 12.6·11-s − 58.3·13-s − 25.2·14-s − 78.9·16-s − 46.0·17-s + 139.·19-s + 56.2·20-s + 45.5·22-s − 23·23-s − 0.564·25-s − 210.·26-s − 35.2·28-s − 247.·29-s + 42.5·31-s − 199.·32-s − 166.·34-s − 78.0·35-s − 42.7·37-s + 502.·38-s − 119.·40-s − 407.·41-s + ⋯ |
| L(s) = 1 | + 1.27·2-s + 0.629·4-s + 0.997·5-s − 0.377·7-s − 0.472·8-s + 1.27·10-s + 0.345·11-s − 1.24·13-s − 0.482·14-s − 1.23·16-s − 0.657·17-s + 1.68·19-s + 0.628·20-s + 0.441·22-s − 0.208·23-s − 0.00451·25-s − 1.58·26-s − 0.238·28-s − 1.58·29-s + 0.246·31-s − 1.10·32-s − 0.839·34-s − 0.377·35-s − 0.190·37-s + 2.14·38-s − 0.471·40-s − 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1449 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 3.61T + 8T^{2} \) |
| 5 | \( 1 - 11.1T + 125T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 247.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 42.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 6.87T + 7.95e4T^{2} \) |
| 47 | \( 1 - 437.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 245.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 747.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 908.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 286.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 628.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 327.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 86.1T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.14e3T + 9.12e5T^{2} \) |
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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
−9.172001010475855974644302926748, −7.65944569075917421223748878890, −6.85283823431838542617994037650, −6.00961030061012287641078468502, −5.37606979679421862503229678381, −4.66990694743014943729049838552, −3.59165895730896587718299920383, −2.74283922085910057441359024884, −1.76293622347007540437628216504, 0,
1.76293622347007540437628216504, 2.74283922085910057441359024884, 3.59165895730896587718299920383, 4.66990694743014943729049838552, 5.37606979679421862503229678381, 6.00961030061012287641078468502, 6.85283823431838542617994037650, 7.65944569075917421223748878890, 9.172001010475855974644302926748
