L(s) = 1 | + (0.719 − 1.24i)2-s + (−0.104 + 0.994i)3-s + (−0.534 − 0.926i)4-s + (0.997 + 1.72i)5-s + (1.16 + 0.845i)6-s − 0.100·8-s + (−0.978 − 0.207i)9-s + 2.87·10-s + (0.374 − 0.648i)11-s + (0.977 − 0.435i)12-s + (−1.82 + 0.811i)15-s + (0.462 − 0.801i)16-s + 0.347·17-s + (−0.962 + 1.06i)18-s + (1.06 − 1.84i)20-s + ⋯ |
L(s) = 1 | + (0.719 − 1.24i)2-s + (−0.104 + 0.994i)3-s + (−0.534 − 0.926i)4-s + (0.997 + 1.72i)5-s + (1.16 + 0.845i)6-s − 0.100·8-s + (−0.978 − 0.207i)9-s + 2.87·10-s + (0.374 − 0.648i)11-s + (0.977 − 0.435i)12-s + (−1.82 + 0.811i)15-s + (0.462 − 0.801i)16-s + 0.347·17-s + (−0.962 + 1.06i)18-s + (1.06 − 1.84i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.973955769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.973955769\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.104 - 0.994i)T \) |
| 239 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.719 + 1.24i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.997 - 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.374 + 0.648i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.848 - 1.46i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.961 + 1.66i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 0.618T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.882 + 1.52i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.716116344506322578271487281829, −9.018072252947797977269237137583, −7.65025163872624006985384437580, −6.70899504128218267763661069985, −5.76238246199488826113256529223, −5.32280337248662310963812374188, −3.96809230738928542715019338230, −3.42805355450789153828449672968, −2.76454031636809279004378509639, −1.85169019732102137321684816579,
1.28513377143557202752403216467, 2.09419898732789647147549954291, 3.89076934471466921933478666850, 4.96334794275635969081449843893, 5.30376313681317690609111543065, 6.12690071607839028743575783672, 6.65515343615949225531721648931, 7.68713755864819826759685226929, 8.154736968134464318346814747265, 9.039648963825575934556290616090