L(s) = 1 | + 1.51·2-s + (−1.66 − 0.476i)3-s + 0.294·4-s + (−0.775 − 2.09i)5-s + (−2.52 − 0.722i)6-s − 2.58·8-s + (2.54 + 1.58i)9-s + (−1.17 − 3.17i)10-s + 2.14i·11-s + (−0.490 − 0.140i)12-s − 3.48·13-s + (0.291 + 3.86i)15-s − 4.50·16-s + 3.57i·17-s + (3.85 + 2.40i)18-s + 1.22i·19-s + ⋯ |
L(s) = 1 | + 1.07·2-s + (−0.961 − 0.275i)3-s + 0.147·4-s + (−0.346 − 0.937i)5-s + (−1.02 − 0.294i)6-s − 0.913·8-s + (0.848 + 0.529i)9-s + (−0.371 − 1.00i)10-s + 0.647i·11-s + (−0.141 − 0.0405i)12-s − 0.965·13-s + (0.0753 + 0.997i)15-s − 1.12·16-s + 0.867i·17-s + (0.908 + 0.566i)18-s + 0.280i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0839366 + 0.203718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0839366 + 0.203718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.66 + 0.476i)T \) |
| 5 | \( 1 + (0.775 + 2.09i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 11 | \( 1 - 2.14iT - 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 - 1.22iT - 19T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 29 | \( 1 - 5.95iT - 29T^{2} \) |
| 31 | \( 1 - 3.17iT - 31T^{2} \) |
| 37 | \( 1 + 7.80iT - 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.99iT - 43T^{2} \) |
| 47 | \( 1 - 6.10iT - 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 - 3.39iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 6.85T + 73T^{2} \) |
| 79 | \( 1 + 1.88T + 79T^{2} \) |
| 83 | \( 1 + 9.10iT - 83T^{2} \) |
| 89 | \( 1 + 1.77T + 89T^{2} \) |
| 97 | \( 1 - 1.32T + 97T^{2} \) |
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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
−10.94028142624124357748740050761, −9.944311125257716208181376651947, −9.043264273616780279083941377979, −7.952627070030462721580826684197, −6.98056512906889931223861710088, −5.98356115136556706648001045116, −5.08241545835508513858864412576, −4.65120781254780243575294469846, −3.61974835770810955237085437236, −1.78971060072471725866905761065,
0.087111637597271660231347698180, 2.69080503484688617720983083181, 3.69472151342253410284996595574, 4.64452426584128968092730749430, 5.42176255155844099094125183464, 6.36995251949364551214816259926, 6.99382279734711269543810997667, 8.184007563182189961248838134310, 9.587373124831260259540590268138, 10.11603395903263836959425352500