| L(s) = 1 | + (−0.819 − 0.573i)2-s + (−0.914 − 1.47i)3-s + (0.342 + 0.939i)4-s + (−2.23 − 0.125i)5-s + (−0.0940 + 1.72i)6-s + (2.86 − 0.768i)7-s + (0.258 − 0.965i)8-s + (−1.32 + 2.69i)9-s + (1.75 + 1.38i)10-s + (−0.0113 + 0.00652i)11-s + (1.06 − 1.36i)12-s + (−6.41 + 0.561i)13-s + (−2.79 − 1.01i)14-s + (1.85 + 3.39i)15-s + (−0.766 + 0.642i)16-s + (−1.95 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.405i)2-s + (−0.528 − 0.849i)3-s + (0.171 + 0.469i)4-s + (−0.998 − 0.0561i)5-s + (−0.0384 + 0.706i)6-s + (1.08 − 0.290i)7-s + (0.0915 − 0.341i)8-s + (−0.441 + 0.897i)9-s + (0.555 + 0.437i)10-s + (−0.00340 + 0.00196i)11-s + (0.308 − 0.393i)12-s + (−1.77 + 0.155i)13-s + (−0.745 − 0.271i)14-s + (0.479 + 0.877i)15-s + (−0.191 + 0.160i)16-s + (−0.473 − 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.229968 + 0.159449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.229968 + 0.159449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 + 0.573i)T \) |
| 3 | \( 1 + (0.914 + 1.47i)T \) |
| 5 | \( 1 + (2.23 + 0.125i)T \) |
| 19 | \( 1 + (3.64 - 2.38i)T \) |
| good | 7 | \( 1 + (-2.86 + 0.768i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (0.0113 - 0.00652i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.41 - 0.561i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (1.95 + 1.36i)T + (5.81 + 15.9i)T^{2} \) |
| 23 | \( 1 + (-4.26 - 1.98i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.635 - 3.60i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (4.33 - 7.51i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.43 - 4.43i)T + 37iT^{2} \) |
| 41 | \( 1 + (-2.50 - 2.98i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-6.75 + 3.15i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.0707 - 0.101i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (6.95 + 3.24i)T + (34.0 + 40.6i)T^{2} \) |
| 59 | \( 1 + (0.528 + 2.99i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.13 + 0.775i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (5.26 - 3.68i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (5.43 - 14.9i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.662 - 7.57i)T + (-71.8 - 12.6i)T^{2} \) |
| 79 | \( 1 + (7.10 + 8.47i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (5.04 - 1.35i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (2.01 + 1.69i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (6.61 - 9.44i)T + (-33.1 - 91.1i)T^{2} \) |
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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
−11.15221406695277350690634374089, −10.30454408563432187016842504729, −8.969729402345826319050821135845, −8.109881861134886498358103228755, −7.40158399428616233229917724225, −6.88558795456780023005575602527, −5.17979645163116484690328149950, −4.41672840818730013375639016814, −2.72734585247068198737115479595, −1.40902846074972111783284379554,
0.21237880286660018763326510239, 2.51844437021957884990855217711, 4.32291388189393930191561701729, 4.80602230767910078322518820953, 5.94146681509550697014908151408, 7.18979902286817875021896222068, 7.893951889656437525306477555057, 8.874350087212468475493976157075, 9.571863417266852876711561619312, 10.80446184251694839347815590618
