| L(s) = 1 | + (−0.0775 + 0.134i)2-s + (−0.124 + 0.707i)3-s + (0.987 + 1.71i)4-s + (−0.326 + 0.118i)5-s + (−0.0853 − 0.0715i)6-s + (−1.80 + 0.657i)7-s − 0.616·8-s + (2.33 + 0.849i)9-s + (0.00935 − 0.0530i)10-s + (−0.257 − 1.45i)11-s + (−1.33 + 0.485i)12-s + (2.58 − 0.941i)13-s + (0.0517 − 0.293i)14-s + (−0.0433 − 0.245i)15-s + (−1.92 + 3.33i)16-s + (1.05 − 1.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.0548 + 0.0949i)2-s + (−0.0719 + 0.408i)3-s + (0.493 + 0.855i)4-s + (−0.145 + 0.0531i)5-s + (−0.0348 − 0.0292i)6-s + (−0.682 + 0.248i)7-s − 0.218·8-s + (0.778 + 0.283i)9-s + (0.00295 − 0.0167i)10-s + (−0.0775 − 0.439i)11-s + (−0.384 + 0.140i)12-s + (0.717 − 0.261i)13-s + (0.0138 − 0.0784i)14-s + (−0.0111 − 0.0634i)15-s + (−0.482 + 0.834i)16-s + (0.255 − 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.895709 + 0.528149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.895709 + 0.528149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-10.2 + 1.91i)T \) |
| good | 2 | \( 1 + (0.0775 - 0.134i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.124 - 0.707i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (0.326 - 0.118i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.80 - 0.657i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.257 + 1.45i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.58 + 0.941i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.05 + 1.82i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.49 + 4.32i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.58 + 7.93i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.605 + 3.43i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.25 - 0.822i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.677 - 0.246i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 + (1.87 - 3.24i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.901 + 0.756i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (3.27 - 1.19i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.63 - 9.27i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-6.88 - 5.77i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.84 + 1.03i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (3.65 + 6.32i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.306 + 1.73i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (10.0 + 3.67i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.40 - 7.99i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (5.19 + 4.36i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-11.5 + 4.18i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.63528499934725675369140027373, −12.82601635312672415856559200327, −11.78118730784647259118363337861, −10.77998937858555679730484599120, −9.606740233421468345741360291181, −8.374329915853000374454123503786, −7.25686643344978605718443548027, −6.06810206288206528711024386816, −4.23663362277313001119392312931, −2.88851623928332402880747055726,
1.58651156064519791079489699005, 3.81705559976217993599236249300, 5.74322528924316342016314037438, 6.69718106135323292639801864550, 7.80908802940066747278193249963, 9.632412685333004387528616751018, 10.14330450878024420336794889776, 11.50318613648390621320834963320, 12.40504889788238682793789414474, 13.50262853119760236612272546976
