L(s) = 1 | + 2.00e3i·3-s − 3.49e4i·5-s + (7.70e5 − 2.90e5i)7-s + 7.57e5·9-s + 8.85e6·11-s + 1.08e8i·13-s + 7.01e7·15-s − 1.83e8i·17-s + 2.90e8i·19-s + (5.83e8 + 1.54e9i)21-s + 1.47e9·23-s − 1.22e9·25-s + 1.11e10i·27-s − 3.38e10·29-s − 1.88e10i·31-s + ⋯ |
L(s) = 1 | + 0.917i·3-s − 0.447i·5-s + (0.935 − 0.352i)7-s + 0.158·9-s + 0.454·11-s + 1.73i·13-s + 0.410·15-s − 0.447i·17-s + 0.324i·19-s + (0.323 + 0.858i)21-s + 0.433·23-s − 0.199·25-s + 1.06i·27-s − 1.96·29-s − 0.686i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{15}{2})\) |
\(\approx\) |
\(2.508918924\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.508918924\) |
\(L(8)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 3.49e4iT \) |
| 7 | \( 1 + (-7.70e5 + 2.90e5i)T \) |
good | 3 | \( 1 - 2.00e3iT - 4.78e6T^{2} \) |
| 11 | \( 1 - 8.85e6T + 3.79e14T^{2} \) |
| 13 | \( 1 - 1.08e8iT - 3.93e15T^{2} \) |
| 17 | \( 1 + 1.83e8iT - 1.68e17T^{2} \) |
| 19 | \( 1 - 2.90e8iT - 7.99e17T^{2} \) |
| 23 | \( 1 - 1.47e9T + 1.15e19T^{2} \) |
| 29 | \( 1 + 3.38e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + 1.88e10iT - 7.56e20T^{2} \) |
| 37 | \( 1 - 1.25e10T + 9.01e21T^{2} \) |
| 41 | \( 1 - 2.75e11iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 3.87e11T + 7.38e22T^{2} \) |
| 47 | \( 1 + 1.28e11iT - 2.56e23T^{2} \) |
| 53 | \( 1 + 1.28e12T + 1.37e24T^{2} \) |
| 59 | \( 1 - 2.40e12iT - 6.19e24T^{2} \) |
| 61 | \( 1 + 4.10e12iT - 9.87e24T^{2} \) |
| 67 | \( 1 - 5.61e12T + 3.67e25T^{2} \) |
| 71 | \( 1 - 1.44e13T + 8.27e25T^{2} \) |
| 73 | \( 1 - 1.79e13iT - 1.22e26T^{2} \) |
| 79 | \( 1 - 1.10e13T + 3.68e26T^{2} \) |
| 83 | \( 1 - 1.83e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 - 7.53e13iT - 1.95e27T^{2} \) |
| 97 | \( 1 + 7.51e13iT - 6.52e27T^{2} \) |
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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
−11.04029016173819519859876628891, −9.593943556292258999089183310159, −9.196568595392218902421480790467, −7.87291522396619374852735231699, −6.79321123561814320947239836865, −5.31818199733332113109710454853, −4.38187564432930654133146937822, −3.85328711826060554124508085542, −2.01936444262484971387280248039, −1.11457583763558004759830528417,
0.49154287454708373861231521453, 1.48839192853267326361288878516, 2.41640335179376418375358617829, 3.69676436600291160790493819272, 5.16398892216659241080084226918, 6.13004499024250253875059532800, 7.35995374969015984369550689147, 7.906768407789014487231901567759, 9.084610400356952461059001688515, 10.47858929760225176566678223436