| L(s) = 1 | + (0.453 − 0.891i)2-s + (0.0523 − 0.998i)3-s + (−0.587 − 0.809i)4-s + (1.98 − 1.02i)5-s + (−0.866 − 0.5i)6-s + (0.742 − 0.916i)7-s + (−0.987 + 0.156i)8-s + (−0.994 − 0.104i)9-s + (−0.00699 − 2.23i)10-s + (0.0138 − 0.0311i)11-s + (−0.838 + 0.544i)12-s + (−4.42 − 2.87i)13-s + (−0.479 − 1.07i)14-s + (−0.915 − 2.03i)15-s + (−0.309 + 0.951i)16-s + (−0.159 + 0.415i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (0.0302 − 0.576i)3-s + (−0.293 − 0.404i)4-s + (0.889 − 0.456i)5-s + (−0.353 − 0.204i)6-s + (0.280 − 0.346i)7-s + (−0.349 + 0.0553i)8-s + (−0.331 − 0.0348i)9-s + (−0.00221 − 0.707i)10-s + (0.00417 − 0.00938i)11-s + (−0.242 + 0.157i)12-s + (−1.22 − 0.796i)13-s + (−0.128 − 0.288i)14-s + (−0.236 − 0.526i)15-s + (−0.0772 + 0.237i)16-s + (−0.0387 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.428747 - 1.82690i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428747 - 1.82690i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.453 + 0.891i)T \) |
| 3 | \( 1 + (-0.0523 + 0.998i)T \) |
| 5 | \( 1 + (-1.98 + 1.02i)T \) |
| 31 | \( 1 + (-5.52 - 0.711i)T \) |
| good | 7 | \( 1 + (-0.742 + 0.916i)T + (-1.45 - 6.84i)T^{2} \) |
| 11 | \( 1 + (-0.0138 + 0.0311i)T + (-7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (4.42 + 2.87i)T + (5.28 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.159 - 0.415i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.333 + 1.56i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (0.233 + 1.47i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (0.808 + 2.48i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (-1.39 - 5.21i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.36 + 3.73i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (4.59 + 7.06i)T + (-17.4 + 39.2i)T^{2} \) |
| 47 | \( 1 + (4.96 - 2.52i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-4.48 + 3.63i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-0.357 + 0.321i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 2.49iT - 61T^{2} \) |
| 67 | \( 1 + (2.12 - 7.91i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.0597 + 0.568i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.466 - 1.21i)T + (-54.2 + 48.8i)T^{2} \) |
| 79 | \( 1 + (-0.906 + 0.403i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (12.6 - 0.662i)T + (82.5 - 8.67i)T^{2} \) |
| 89 | \( 1 + (-8.03 + 5.83i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-13.8 - 2.19i)T + (92.2 + 29.9i)T^{2} \) |
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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
−9.969636654383507139551274240791, −8.965763979057901561703104879504, −8.110310250960681169285822868981, −7.12463340883119541593167686952, −6.10942471505608806695549527928, −5.22667546501988072187088162830, −4.47199571252726956289401585005, −2.93888742539168888340745553675, −2.06657918393384814736075054197, −0.78458177412125361527038014834,
2.07801430628806200985437092672, 3.15419743550234512302824740006, 4.47211536332714428781160555828, 5.19706146410378331936823325768, 6.06459623286165614149421953076, 6.89979993235365870540694544412, 7.80389056074554705253831949275, 8.869281920297538103756832192777, 9.591536229543373243014580039040, 10.15268350970235468471986776986
