L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (2 + 1.73i)7-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)11-s + 2·13-s + 0.999·15-s + (−0.5 − 0.866i)17-s + (1.5 − 2.59i)19-s + (−0.499 + 2.59i)21-s + (2.5 − 4.33i)23-s + (2 + 3.46i)25-s + 5·27-s − 2·29-s + (−2.5 − 4.33i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + (0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s + 0.554·13-s + 0.258·15-s + (−0.121 − 0.210i)17-s + (0.344 − 0.596i)19-s + (−0.109 + 0.566i)21-s + (0.521 − 0.902i)23-s + (0.400 + 0.692i)25-s + 0.962·27-s − 0.371·29-s + (−0.449 − 0.777i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.121520170\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.121520170\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (2.5 - 4.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.583494187759943191617791722417, −8.899752144273314320040223909686, −8.482352520140639186940626269576, −7.39858090564515469227861968815, −6.31836680348769194072082136327, −5.39258882848160249493338012395, −4.68075855893167676331833226928, −3.57165973902465274383172110991, −2.54398730826856149381154172190, −1.06437526557409089207291463670,
1.39101956254902179472605731523, 2.25941365632885050281541916607, 3.59820411267085500824298567338, 4.66216835570596692595637984487, 5.51213348401063149214960519876, 6.74128116068101919904724702578, 7.44021761200527431332599522526, 7.937867437248048029814234220065, 8.899485584939969874396445523537, 9.961008004776927866165771651760