L(s) = 1 | + (−0.235 − 0.407i)2-s + (−1.12 + 1.94i)3-s + (0.889 − 1.54i)4-s + (−0.264 − 0.458i)5-s + 1.05·6-s − 1.77·8-s + (−1.02 − 1.78i)9-s + (−0.124 + 0.215i)10-s + (1.12 − 1.94i)11-s + (2 + 3.46i)12-s + 13-s + 1.19·15-s + (−1.35 − 2.35i)16-s + (0.653 − 1.13i)17-s + (−0.484 + 0.839i)18-s + (0.735 + 1.27i)19-s + ⋯ |
L(s) = 1 | + (−0.166 − 0.288i)2-s + (−0.649 + 1.12i)3-s + (0.444 − 0.770i)4-s + (−0.118 − 0.205i)5-s + 0.432·6-s − 0.628·8-s + (−0.343 − 0.594i)9-s + (−0.0393 + 0.0682i)10-s + (0.339 − 0.587i)11-s + (0.577 + 0.999i)12-s + 0.277·13-s + 0.307·15-s + (−0.339 − 0.588i)16-s + (0.158 − 0.274i)17-s + (−0.114 + 0.197i)18-s + (0.168 + 0.292i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.986418 - 0.488994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.986418 - 0.488994i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (0.235 + 0.407i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.12 - 1.94i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.264 + 0.458i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.12 + 1.94i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.653 + 1.13i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.735 - 1.27i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.91 + 5.05i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 + (-3.51 + 6.08i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.18 - 2.04i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.49T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + (4.29 + 7.43i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.63 - 9.76i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.08 + 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.96 + 13.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + (3.82 - 6.61i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.669 - 1.15i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 + (3.45 + 5.98i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−10.43970087174889203232044648567, −9.865854616790587797705492040034, −9.044462578526515083663414115452, −7.998230508727277926421203902083, −6.48400325515701686641066707702, −5.90366341639644609967063000181, −4.89507096451576081807615852798, −4.02051541233478421603970015196, −2.57692686102129289810564038069, −0.75230714858177226726302529098,
1.39682197570714546692433927274, 2.80551532454422095122997885018, 4.08616416955677113954172633773, 5.62850828515683812886938397045, 6.50036349460008867071216105659, 7.15483767952491840112803450293, 7.74682007683468155012351747310, 8.720473208440529915708843276356, 9.794251117993925509644333827661, 11.10746955412724924006116828589