L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.150 − 2.23i)5-s + (−2.28 + 1.32i)7-s − 0.999·8-s + (−1.85 − 1.24i)10-s + (2.86 + 1.65i)11-s + (2.91 + 5.05i)13-s + (0.00309 + 2.64i)14-s + (−0.5 + 0.866i)16-s + 2.78i·17-s − 0.0210i·19-s + (−2.00 + 0.984i)20-s + (2.86 − 1.65i)22-s + (2.80 + 4.85i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.0674 − 0.997i)5-s + (−0.865 + 0.501i)7-s − 0.353·8-s + (−0.587 − 0.394i)10-s + (0.865 + 0.499i)11-s + (0.809 + 1.40i)13-s + (0.000827 + 0.707i)14-s + (−0.125 + 0.216i)16-s + 0.674i·17-s − 0.00483i·19-s + (−0.448 + 0.220i)20-s + (0.611 − 0.353i)22-s + (0.583 + 1.01i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.775338592\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.775338592\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.150 + 2.23i)T \) |
| 7 | \( 1 + (2.28 - 1.32i)T \) |
good | 11 | \( 1 + (-2.86 - 1.65i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.91 - 5.05i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2.78iT - 17T^{2} \) |
| 19 | \( 1 + 0.0210iT - 19T^{2} \) |
| 23 | \( 1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.43 + 0.828i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.24 - 3.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.54iT - 37T^{2} \) |
| 41 | \( 1 + (-5.58 - 9.68i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.13 - 1.23i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.89 - 2.24i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.154T + 53T^{2} \) |
| 59 | \( 1 + (-3.32 - 5.75i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.58 + 4.38i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.9 + 6.89i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 + 0.567T + 73T^{2} \) |
| 79 | \( 1 + (-2.02 + 3.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.5 - 6.66i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.33T + 89T^{2} \) |
| 97 | \( 1 + (4.93 - 8.54i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.098416248844371778587495144337, −9.027309630336741359730061747691, −7.70843980456914660838455553007, −6.55041181796231786598139110190, −6.01502323927314333308882039324, −5.04499038067542626507974179787, −4.09888413525028463660529207659, −3.54535784622866820297161810329, −2.07568894121087231111937122758, −1.25382922013063988574152667673,
0.64227433289911072051500375950, 2.66607633168060673113692153531, 3.45313156785415977680135655197, 4.05950893138784620222284777421, 5.50925325211850187470477878800, 6.07031927370294931612387067538, 6.87662053443733573260929496738, 7.34309026324695458808547575435, 8.351071061088146651696900725202, 9.148822277233991292358345722467