L(s) = 1 | + (−0.866 + 0.5i)5-s + (0.5 − 0.866i)7-s + (−2.59 − 1.5i)11-s + (4.58 − 2.64i)13-s − 5.29·17-s + 5.29i·19-s + (−2.64 − 4.58i)23-s + (−2 + 3.46i)25-s + (5.19 + 3i)29-s + (−3.5 − 6.06i)31-s + 0.999i·35-s + 5.29i·37-s + (−2.64 − 4.58i)41-s + (−9.16 − 5.29i)43-s + (3 + 5.19i)49-s + ⋯ |
L(s) = 1 | + (−0.387 + 0.223i)5-s + (0.188 − 0.327i)7-s + (−0.783 − 0.452i)11-s + (1.27 − 0.733i)13-s − 1.28·17-s + 1.21i·19-s + (−0.551 − 0.955i)23-s + (−0.400 + 0.692i)25-s + (0.964 + 0.557i)29-s + (−0.628 − 1.08i)31-s + 0.169i·35-s + 0.869i·37-s + (−0.413 − 0.715i)41-s + (−1.39 − 0.806i)43-s + (0.428 + 0.742i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09657754691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09657754691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.866 - 0.5i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.59 + 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.58 + 2.64i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 - 5.29iT - 19T^{2} \) |
| 23 | \( 1 + (2.64 + 4.58i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.19 - 3i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.29iT - 37T^{2} \) |
| 41 | \( 1 + (2.64 + 4.58i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (9.16 + 5.29i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + (3.46 - 2i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 3T + 73T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.06 - 3.5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.315765901382988526307521511667, −7.951043435695113757201322911154, −6.98892221220991029353975411524, −6.13878341316070335992822643076, −5.49487274942091809875671570692, −4.37739407692341192815059591326, −3.66778831681467163213491640747, −2.76220234475831489304915850563, −1.49257720046064312244207441231, −0.03091430045308278254473030860,
1.62060260981156723387388775869, 2.57877846619141349489150269374, 3.74078240587203922901239818808, 4.56039458289133017492077353642, 5.23046618634070937203282247815, 6.33716851995500793202092297073, 6.87339164117732663715866484812, 7.88309047387293185000115232021, 8.544740889078321305962215661124, 9.064768310878275141742170303272