L(s) = 1 | + 3i·3-s + 15.6·5-s + (−10.8 − 15.0i)7-s − 9·9-s − 41.3·11-s + 8.02·13-s + 47.0i·15-s − 14.2i·17-s − 10.2i·19-s + (45.0 − 32.5i)21-s + 203. i·23-s + 121.·25-s − 27i·27-s − 82.6i·29-s + 156.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.40·5-s + (−0.585 − 0.810i)7-s − 0.333·9-s − 1.13·11-s + 0.171·13-s + 0.810i·15-s − 0.202i·17-s − 0.123i·19-s + (0.468 − 0.337i)21-s + 1.84i·23-s + 0.969·25-s − 0.192i·27-s − 0.529i·29-s + 0.905·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7250726619\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7250726619\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (10.8 + 15.0i)T \) |
good | 5 | \( 1 - 15.6T + 125T^{2} \) |
| 11 | \( 1 + 41.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 8.02T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 10.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 203. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 82.6iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 106. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 315. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 474.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 266. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 425. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 8.30T + 2.26e5T^{2} \) |
| 67 | \( 1 + 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 109. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 53.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 507. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.06e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 573. iT - 9.12e5T^{2} \) |
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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
−9.331452833751330975318611418097, −8.175735598033670154699954266337, −7.28673929233823978939577911009, −6.32054400885132528513477052046, −5.57057748746664058737242645942, −4.86921114183428745928145428840, −3.65095284703160781629904822336, −2.76153947964473270581030131331, −1.62133613807346604380769148486, −0.14991983195575742606897298036,
1.37202680962342255365203954661, 2.47361702723809010845768584148, 2.92540609942541689406859855895, 4.71755886842369945271122425326, 5.57283646577125908070005613638, 6.24640896138433057790971180257, 6.78689794591056815718675273680, 8.118809616543252204238885326193, 8.636973174311779024368240725447, 9.620184974304895416674072180530