L(s) = 1 | + (3.10e4 − 9.74e4i)3-s + 2.82e7i·5-s + 1.02e8i·7-s + (−8.53e9 − 6.04e9i)9-s + 1.32e11·11-s − 9.58e11·13-s + (2.75e12 + 8.76e11i)15-s + 2.11e12i·17-s − 5.68e12i·19-s + (1.00e13 + 3.19e12i)21-s − 9.10e13·23-s − 3.22e14·25-s + (−8.53e14 + 6.44e14i)27-s − 3.40e15i·29-s − 2.15e15i·31-s + ⋯ |
L(s) = 1 | + (0.303 − 0.952i)3-s + 1.29i·5-s + 0.137i·7-s + (−0.816 − 0.577i)9-s + 1.54·11-s − 1.92·13-s + (1.23 + 0.392i)15-s + 0.254i·17-s − 0.212i·19-s + (0.131 + 0.0417i)21-s − 0.458·23-s − 0.676·25-s + (−0.797 + 0.602i)27-s − 1.50i·29-s − 0.471i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.738i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.673 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(2.051341696\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.051341696\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.10e4 + 9.74e4i)T \) |
good | 5 | \( 1 - 2.82e7iT - 4.76e14T^{2} \) |
| 7 | \( 1 - 1.02e8iT - 5.58e17T^{2} \) |
| 11 | \( 1 - 1.32e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 9.58e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.11e12iT - 6.90e25T^{2} \) |
| 19 | \( 1 + 5.68e12iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 9.10e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.40e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 2.15e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 + 2.84e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.99e16iT - 7.38e33T^{2} \) |
| 43 | \( 1 + 6.56e16iT - 2.00e34T^{2} \) |
| 47 | \( 1 - 6.12e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.81e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 - 4.67e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 6.14e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.06e19iT - 2.22e38T^{2} \) |
| 71 | \( 1 + 2.58e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.97e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.51e20iT - 7.08e39T^{2} \) |
| 83 | \( 1 - 1.89e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.33e20iT - 8.65e40T^{2} \) |
| 97 | \( 1 - 4.64e20T + 5.27e41T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.65074099666962021911395351970, −10.16355540986945712180428527230, −9.031805214902551600229499849116, −7.53072502642904980915773131744, −6.90052997637886873602638584932, −5.91426442033130811941436265491, −4.02266880597164504742772221403, −2.72666140934624804211960570265, −2.01474391722111379335557940914, −0.52025155976046779624047414931,
0.76602067353757326257219987909, 2.09642423814599634725175904333, 3.63308206313989874065868512137, 4.62259623471215877462285361377, 5.37076811289125214635596850440, 7.13453952274186370157206620195, 8.617863400541739287861952882530, 9.279013358549163866278698047113, 10.20500642498547520508336639230, 11.77609973407352939264881677659