L(s) = 1 | + (−1.72 − 0.158i)3-s + (0.5 + 0.866i)5-s + (−0.724 + 1.25i)7-s + (2.94 + 0.548i)9-s + (−0.724 − 1.57i)15-s − 2·17-s − 2.89·19-s + (1.44 − 2.04i)21-s + (−1.27 − 2.20i)23-s + (−0.499 + 0.866i)25-s + (−4.99 − 1.41i)27-s + (−3.94 + 6.84i)29-s + (5.44 + 9.43i)31-s − 1.44·35-s − 6·37-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.0917i)3-s + (0.223 + 0.387i)5-s + (−0.273 + 0.474i)7-s + (0.983 + 0.182i)9-s + (−0.187 − 0.406i)15-s − 0.485·17-s − 0.665·19-s + (0.316 − 0.447i)21-s + (−0.265 − 0.460i)23-s + (−0.0999 + 0.173i)25-s + (−0.962 − 0.272i)27-s + (−0.733 + 1.27i)29-s + (0.978 + 1.69i)31-s − 0.245·35-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.251814 + 0.533576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251814 + 0.533576i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.158i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 7 | \( 1 + (0.724 - 1.25i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + (1.27 + 2.20i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.94 - 6.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.44 - 9.43i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + (-0.0505 - 0.0874i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.89 - 6.75i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.27 - 3.94i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + (5.44 + 9.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.62 - 9.74i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + (-1.44 + 2.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.275 - 0.476i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.78832265312632854543081591459, −10.04934859647792007096248730385, −9.118734577754998923965232540623, −8.105113320491477142605880916393, −6.83317749925184454572522298390, −6.44333503832701885315078760798, −5.39376855765798029680332070095, −4.53139349427312735920576175321, −3.12756383053117930732439469251, −1.68178210662869151364463014952,
0.34073225336231518616438204083, 1.97040975884848632450851485273, 3.81841909508918004462768781279, 4.62183148940079457118298043420, 5.69454031474100674333405374482, 6.41904495939661558822389766874, 7.33585201376107346540192324198, 8.360538815743739287825645014039, 9.544078160972602270924236149454, 10.07599742342496039545376373245