L(s) = 1 | + (−0.333 − 0.578i)2-s + (−0.5 − 0.866i)3-s + (0.777 − 1.34i)4-s + 0.849·5-s + (−0.333 + 0.578i)6-s + (1.61 − 2.79i)7-s − 2.37·8-s + (−0.499 + 0.866i)9-s + (−0.283 − 0.490i)10-s + (0.5 + 0.866i)11-s − 1.55·12-s + (2.86 − 2.18i)13-s − 2.15·14-s + (−0.424 − 0.735i)15-s + (−0.762 − 1.32i)16-s + (0.394 − 0.683i)17-s + ⋯ |
L(s) = 1 | + (−0.236 − 0.408i)2-s + (−0.288 − 0.499i)3-s + (0.388 − 0.673i)4-s + 0.379·5-s + (−0.136 + 0.236i)6-s + (0.610 − 1.05i)7-s − 0.838·8-s + (−0.166 + 0.288i)9-s + (−0.0896 − 0.155i)10-s + (0.150 + 0.261i)11-s − 0.448·12-s + (0.795 − 0.606i)13-s − 0.576·14-s + (−0.109 − 0.189i)15-s + (−0.190 − 0.330i)16-s + (0.0956 − 0.165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.581867 - 1.17475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.581867 - 1.17475i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-2.86 + 2.18i)T \) |
good | 2 | \( 1 + (0.333 + 0.578i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 0.849T + 5T^{2} \) |
| 7 | \( 1 + (-1.61 + 2.79i)T + (-3.5 - 6.06i)T^{2} \) |
| 17 | \( 1 + (-0.394 + 0.683i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.49 - 2.58i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.75 - 3.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.576 + 0.998i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.0322T + 31T^{2} \) |
| 37 | \( 1 + (2.53 + 4.38i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.86 + 3.23i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.10 + 1.90i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.56T + 47T^{2} \) |
| 53 | \( 1 - 0.836T + 53T^{2} \) |
| 59 | \( 1 + (4.95 - 8.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.25 + 5.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.97 - 8.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.15 - 3.72i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + (-4.75 - 8.23i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.84 - 10.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.77588748700267947709951430275, −10.25353599950128583462880996499, −9.223592346428917925626259223051, −7.991265617932282408215125981505, −7.10375777084580595468734885463, −6.09260860545516295761545757440, −5.26586307239486690266996256371, −3.74816904162277464183065581411, −2.04546535272199327544069989774, −0.986903610854775665953142493778,
2.14038996312803542883191918200, 3.48523964009994804775956055697, 4.83662692503042409393283588191, 5.98866647145829819409929284765, 6.62961578152440419263512139428, 8.020091625956454254578532449506, 8.751474534902907363270586520075, 9.380709855011164389711919105600, 10.74979165104961578046266872621, 11.49829363478726309086662593340