L(s) = 1 | − 2.51i·2-s + i·3-s − 4.34·4-s + (−2.17 − 0.508i)5-s + 2.51·6-s − i·7-s + 5.89i·8-s − 9-s + (−1.28 + 5.48i)10-s − 11-s − 4.34i·12-s − 1.93i·13-s − 2.51·14-s + (0.508 − 2.17i)15-s + 6.17·16-s + 2.73i·17-s + ⋯ |
L(s) = 1 | − 1.78i·2-s + 0.577i·3-s − 2.17·4-s + (−0.973 − 0.227i)5-s + 1.02·6-s − 0.377i·7-s + 2.08i·8-s − 0.333·9-s + (−0.405 + 1.73i)10-s − 0.301·11-s − 1.25i·12-s − 0.535i·13-s − 0.673·14-s + (0.131 − 0.562i)15-s + 1.54·16-s + 0.663i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5946549721\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5946549721\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.17 + 0.508i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.51iT - 2T^{2} \) |
| 13 | \( 1 + 1.93iT - 13T^{2} \) |
| 17 | \( 1 - 2.73iT - 17T^{2} \) |
| 19 | \( 1 + 5.74T + 19T^{2} \) |
| 23 | \( 1 - 3.98iT - 23T^{2} \) |
| 29 | \( 1 - 0.897T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 3.09iT - 37T^{2} \) |
| 41 | \( 1 - 7.83T + 41T^{2} \) |
| 43 | \( 1 - 4.01iT - 43T^{2} \) |
| 47 | \( 1 - 3.41iT - 47T^{2} \) |
| 53 | \( 1 - 9.17iT - 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 - 1.57T + 61T^{2} \) |
| 67 | \( 1 + 5.66iT - 67T^{2} \) |
| 71 | \( 1 + 5.57T + 71T^{2} \) |
| 73 | \( 1 + 9.45iT - 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 - 8.24iT - 83T^{2} \) |
| 89 | \( 1 + 9.21T + 89T^{2} \) |
| 97 | \( 1 - 15.8iT - 97T^{2} \) |
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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
−10.04467310675595027236608942031, −9.222718517006327986156685709608, −8.371419687694270805353132318181, −7.77610188464877293798878816330, −6.20388924963629299890072585719, −4.81271378532524994161340801916, −4.30854087661665480130243262281, −3.48726459859363373476052943617, −2.62425080374941864198241866135, −1.07846594233262000067809236127,
0.31661409569929309541003264125, 2.61054529769220567061059917990, 4.14239615057523951221644260483, 4.80799867775855498992941571878, 5.92913537611718014788407935269, 6.68872552575695969352712745657, 7.17550827805322147277272551168, 8.185811252011892771063112121173, 8.431954205613351910675100777133, 9.352811992492193332404458039341