L(s) = 1 | + (−0.996 + 0.0871i)2-s + (1.70 + 0.794i)3-s + (0.984 − 0.173i)4-s + (−1.76 − 0.642i)6-s + (−1.00 + 3.76i)7-s + (−0.965 + 0.258i)8-s + (0.342 + 0.407i)9-s + (−1.56 + 2.71i)11-s + (1.81 + 0.486i)12-s + (−0.684 − 1.46i)13-s + (0.676 − 3.83i)14-s + (0.939 − 0.342i)16-s + (0.551 + 6.30i)17-s + (−0.376 − 0.376i)18-s + (−2.70 − 3.41i)19-s + ⋯ |
L(s) = 1 | + (−0.704 + 0.0616i)2-s + (0.983 + 0.458i)3-s + (0.492 − 0.0868i)4-s + (−0.720 − 0.262i)6-s + (−0.380 + 1.42i)7-s + (−0.341 + 0.0915i)8-s + (0.114 + 0.135i)9-s + (−0.472 + 0.817i)11-s + (0.524 + 0.140i)12-s + (−0.189 − 0.407i)13-s + (0.180 − 1.02i)14-s + (0.234 − 0.0855i)16-s + (0.133 + 1.52i)17-s + (−0.0886 − 0.0886i)18-s + (−0.621 − 0.783i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.345166 + 0.972457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.345166 + 0.972457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.996 - 0.0871i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.70 + 3.41i)T \) |
good | 3 | \( 1 + (-1.70 - 0.794i)T + (1.92 + 2.29i)T^{2} \) |
| 7 | \( 1 + (1.00 - 3.76i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.56 - 2.71i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.684 + 1.46i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-0.551 - 6.30i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (0.0240 + 0.0168i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.496 - 0.416i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (6.55 - 3.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.88 - 4.88i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.29 + 6.29i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-2.28 - 3.26i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-13.3 - 1.16i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (0.168 - 0.241i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (-5.17 - 4.34i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.0829 + 0.470i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.685 - 7.83i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (5.21 + 0.918i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.70 - 5.79i)T + (-46.9 - 55.9i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 4.10i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.7 - 3.42i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-3.79 - 1.38i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.38 - 0.733i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.19113650773606021611897452785, −9.276269225942608238601124653038, −8.804909511548446165609310954290, −8.228008356260421952247321726679, −7.20395414364693961880371106780, −6.12856505650284483712243641542, −5.26098170810974744519363148796, −3.86253235113494583100514405222, −2.76638938327797270890699366387, −2.05109146601034504145083267031,
0.49797612900991508652996897171, 2.00936413807862389325183586982, 3.10090518997749377281539933998, 3.97591596312796050122576768175, 5.46981335889074595381818370413, 6.74276466635373832987931715648, 7.45695531736236825014762857888, 7.914668174254327169937425837538, 8.916031834608351202344786399785, 9.525196961277611809016880534916