L(s) = 1 | + (−1.91 − 2.08i)2-s + (−2.09 + 4.75i)3-s + (−0.661 + 7.97i)4-s − 5i·5-s + (13.9 − 4.75i)6-s − 32.2i·7-s + (17.8 − 13.8i)8-s + (−18.2 − 19.9i)9-s + (−10.4 + 9.57i)10-s − 32.4·11-s + (−36.5 − 19.8i)12-s + 44.9·13-s + (−67.1 + 61.8i)14-s + (23.7 + 10.4i)15-s + (−63.1 − 10.5i)16-s − 79.2i·17-s + ⋯ |
L(s) = 1 | + (−0.677 − 0.735i)2-s + (−0.402 + 0.915i)3-s + (−0.0826 + 0.996i)4-s − 0.447i·5-s + (0.946 − 0.323i)6-s − 1.74i·7-s + (0.789 − 0.614i)8-s + (−0.675 − 0.737i)9-s + (−0.329 + 0.302i)10-s − 0.890·11-s + (−0.878 − 0.477i)12-s + 0.958·13-s + (−1.28 + 1.18i)14-s + (0.409 + 0.180i)15-s + (−0.986 − 0.164i)16-s − 1.13i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.322148 - 0.541379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.322148 - 0.541379i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.91 + 2.08i)T \) |
| 3 | \( 1 + (2.09 - 4.75i)T \) |
| 5 | \( 1 + 5iT \) |
good | 7 | \( 1 + 32.2iT - 343T^{2} \) |
| 11 | \( 1 + 32.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 69.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 72.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 28.4iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 104. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 96.6iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 103.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 544. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 431.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 183.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 290. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 489.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 292.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 656. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 392.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 336. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−13.94271921394241306585105005666, −13.02442717960122676976498695155, −11.47267478261517980145416372817, −10.67294763122637191903891507258, −9.860504601995168625890885991107, −8.576909235900470800417883050271, −7.12833049823628178113077820601, −4.80539587133545961564556079323, −3.53448032692135581987123339336, −0.57694670491373871785808959614,
2.08996216049576441110993098556, 5.65811230040774957286953814073, 6.20892748665712866173078099919, 7.88770179033286336359070648945, 8.643441155235087150349488275028, 10.31630064890632844083083313231, 11.51033499176809148751460718394, 12.70863326464700223549675703042, 13.98633540555544774412765873929, 15.21972844607844658264225045452