| L(s) = 1 | + (2.52 − 1.83i)2-s + (−6.87 + 21.1i)4-s + (−23.3 − 16.9i)5-s + (27.3 − 84.2i)7-s + (52.3 + 161. i)8-s − 90.3·10-s + (−212. − 340. i)11-s + (92.0 − 66.8i)13-s + (−85.4 − 263. i)14-s + (−147. − 107. i)16-s + (−1.04e3 − 756. i)17-s + (−750. − 2.31e3i)19-s + (520. − 377. i)20-s + (−1.16e3 − 470. i)22-s + 1.71e3·23-s + ⋯ |
| L(s) = 1 | + (0.446 − 0.324i)2-s + (−0.214 + 0.661i)4-s + (−0.418 − 0.304i)5-s + (0.211 − 0.649i)7-s + (0.289 + 0.890i)8-s − 0.285·10-s + (−0.529 − 0.848i)11-s + (0.151 − 0.109i)13-s + (−0.116 − 0.358i)14-s + (−0.144 − 0.104i)16-s + (−0.873 − 0.634i)17-s + (−0.477 − 1.46i)19-s + (0.290 − 0.211i)20-s + (−0.511 − 0.207i)22-s + 0.674·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.451892 - 1.00079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451892 - 1.00079i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (212. + 340. i)T \) |
| good | 2 | \( 1 + (-2.52 + 1.83i)T + (9.88 - 30.4i)T^{2} \) |
| 5 | \( 1 + (23.3 + 16.9i)T + (965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (-27.3 + 84.2i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-92.0 + 66.8i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (1.04e3 + 756. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (750. + 2.31e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 1.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-485. + 1.49e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (7.25e3 - 5.27e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-3.91e3 + 1.20e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-5.31e3 - 1.63e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 6.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.43e3 + 1.36e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (4.54e3 - 3.30e3i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.56e3 + 1.09e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.49e3 - 1.08e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 3.34e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-3.55e4 - 2.58e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.74e3 - 1.45e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (7.56e4 - 5.49e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.33e4 - 1.69e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 - 3.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-9.79e4 + 7.11e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−12.78948104905430595811588226926, −11.39502836896106209081115930817, −10.90467156495529790905688961752, −9.051869195608595189971558724885, −8.148449366866114928712142187684, −6.99797053644404952692545086811, −5.08311588078971266065357422929, −4.05907849602327927409342001290, −2.68861513399010444503455791280, −0.36253267686757176462946942269,
1.87119263438346437976537095731, 3.93054394988805960177224230147, 5.19951998256449507218486197649, 6.31956819324158436239148410278, 7.59158004359530821342665253952, 8.977885349375047434185352729230, 10.17266114689089680959165603403, 11.15400267580577832952176489155, 12.49746076975279932548939812921, 13.32631447509898577909120933717
