L(s) = 1 | + 2-s + (−0.707 + 1.22i)3-s − 4-s + (1.34 − 2.32i)5-s + (−0.707 + 1.22i)6-s − 3·8-s + (0.500 + 0.866i)9-s + (1.34 − 2.32i)10-s + (−2.89 + 5.01i)11-s + (0.707 − 1.22i)12-s + (2.75 + 2.32i)13-s + (1.89 + 3.28i)15-s − 16-s + 5.51·17-s + (0.500 + 0.866i)18-s + (1.41 + 2.44i)19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.408 + 0.707i)3-s − 0.5·4-s + (0.600 − 1.03i)5-s + (−0.288 + 0.499i)6-s − 1.06·8-s + (0.166 + 0.288i)9-s + (0.424 − 0.735i)10-s + (−0.873 + 1.51i)11-s + (0.204 − 0.353i)12-s + (0.764 + 0.644i)13-s + (0.490 + 0.848i)15-s − 0.250·16-s + 1.33·17-s + (0.117 + 0.204i)18-s + (0.324 + 0.561i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23725 + 0.977971i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23725 + 0.977971i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.75 - 2.32i)T \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.34 + 2.32i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.89 - 5.01i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 5.51T + 17T^{2} \) |
| 19 | \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 + (-4.39 - 7.61i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.79T + 37T^{2} \) |
| 41 | \( 1 + (4.87 + 8.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.897 + 1.55i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.29 + 5.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 + (0.779 + 1.34i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.89 - 5.01i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.90 - 5.02i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.89 + 10.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + (-2.12 + 3.67i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.49239166599129914673465808447, −9.967692652074767152227603553374, −9.192876948530416544520137210313, −8.351247033531228125959615865214, −7.11589709759761028009772883488, −5.64827132023306636168237167942, −5.12235509072512179965199004612, −4.59609480584769795615810620943, −3.47666847062815129371261462067, −1.63437641373743660032483214025,
0.789510091918761586153806048591, 2.89706855548721225433193763047, 3.47495647249182616856399331900, 5.13393294299939484461198862017, 6.02519488168574611549779174451, 6.33076216481192078467837663356, 7.67406441971494128090932492963, 8.509817418974811368871136409680, 9.714945560175529536453200131274, 10.47813705115508322413368105247