L(s) = 1 | + 2.61·2-s + (1.30 − 2.26i)3-s + 4.85·4-s + (−1.30 + 2.26i)5-s + (3.42 − 5.93i)6-s + 7.47·8-s + (−1.92 − 3.33i)9-s + (−3.42 + 5.93i)10-s + (−0.927 + 1.60i)11-s + (6.35 − 11.0i)12-s + (−2.5 − 2.59i)13-s + (3.42 + 5.93i)15-s + 9.85·16-s − 1.47·17-s + (−5.04 − 8.73i)18-s + (0.927 + 1.60i)19-s + ⋯ |
L(s) = 1 | + 1.85·2-s + (0.755 − 1.30i)3-s + 2.42·4-s + (−0.585 + 1.01i)5-s + (1.39 − 2.42i)6-s + 2.64·8-s + (−0.642 − 1.11i)9-s + (−1.08 + 1.87i)10-s + (−0.279 + 0.484i)11-s + (1.83 − 3.17i)12-s + (−0.693 − 0.720i)13-s + (0.884 + 1.53i)15-s + 2.46·16-s − 0.357·17-s + (−1.18 − 2.05i)18-s + (0.212 + 0.368i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.55554 - 1.56846i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.55554 - 1.56846i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 + (-1.30 + 2.26i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.30 - 2.26i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.927 - 1.60i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + (-0.927 - 1.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.35 - 4.07i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (0.381 + 0.661i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.11 + 1.93i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.88 + 3.25i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.23T + 59T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.35 + 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 - 4.90T + 89T^{2} \) |
| 97 | \( 1 + (9.42 - 16.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.94414540142417443506037448172, −9.895930084054878414588431388709, −7.983638427302731415471483714642, −7.62349574484209797396147154989, −6.79252910823150672822878048793, −6.13898392007460558667063029527, −4.86545141379106381188143809306, −3.62992044980620075378954273268, −2.81402447053182029100347526799, −2.03914034028436336896142654439,
2.32351608880879227192380520532, 3.47515071161759657347342510798, 4.21395319475268831688290622089, 4.79403713733497976918580727950, 5.57029944182178105596647264099, 6.91256867215572089599375996086, 8.053790102898292210512235183565, 8.940124340853274303994246243647, 9.911369940519728172514554061191, 10.94433054022492507375361283342