L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.16 − 1.59i)3-s + (0.809 − 0.587i)4-s + (0.453 − 0.891i)5-s + (1.59 + 1.16i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.896 + 2.76i)9-s + (−0.156 + 0.987i)10-s + (−1.87 − 0.610i)12-s + (−0.297 − 0.0966i)13-s + (0.309 + 0.951i)14-s + (−1.95 + 0.309i)15-s + (0.309 − 0.951i)16-s − 2.90i·18-s + (−1.14 − 0.831i)19-s + ⋯ |
L(s) = 1 | + (−0.951 + 0.309i)2-s + (−1.16 − 1.59i)3-s + (0.809 − 0.587i)4-s + (0.453 − 0.891i)5-s + (1.59 + 1.16i)6-s − i·7-s + (−0.587 + 0.809i)8-s + (−0.896 + 2.76i)9-s + (−0.156 + 0.987i)10-s + (−1.87 − 0.610i)12-s + (−0.297 − 0.0966i)13-s + (0.309 + 0.951i)14-s + (−1.95 + 0.309i)15-s + (0.309 − 0.951i)16-s − 2.90i·18-s + (−1.14 − 0.831i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2888279911\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2888279911\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (1.16 + 1.59i)T + (-0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.297 + 0.0966i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.550 - 1.69i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.533 - 0.734i)T + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.951i)T^{2} \) |
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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
−9.159953744911655557939868526797, −8.135088661262604289624608088787, −7.72046013415697751984557947773, −6.80881436733102621876858989445, −6.29156176424379872643077555277, −5.47449460511366376868331451580, −4.60472924490394064749791934905, −2.30080141689865138661017673037, −1.45499583956826926060293213301, −0.36957256646041690495853063504,
2.17091458178505453995293788491, 3.28949455374398628859810863879, 4.19216960136911368666699894731, 5.45563783701951241810025982874, 6.16845833438832220551385804788, 6.65817683652675841058012926644, 8.130395880347899286488632344400, 8.989864871564318239008921833687, 9.743727817053203826658852297802, 10.12230641232092547735629902502