L(s) = 1 | − 4.62i·3-s − 17.8i·5-s − 13.8·7-s + 5.59·9-s − 2.18i·11-s − 46.3i·13-s − 82.7·15-s − 18.6·17-s + 122. i·19-s + 64.1i·21-s − 134.·23-s − 194.·25-s − 150. i·27-s − 84.5i·29-s − 31.5·31-s + ⋯ |
L(s) = 1 | − 0.890i·3-s − 1.59i·5-s − 0.749·7-s + 0.207·9-s − 0.0599i·11-s − 0.989i·13-s − 1.42·15-s − 0.266·17-s + 1.47i·19-s + 0.667i·21-s − 1.21·23-s − 1.55·25-s − 1.07i·27-s − 0.541i·29-s − 0.182·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4159824014\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4159824014\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 4.62iT - 27T^{2} \) |
| 5 | \( 1 + 17.8iT - 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 11 | \( 1 + 2.18iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 46.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 18.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 84.5iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 31.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 126. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 168. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 182.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 37.0iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 624. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 246. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 129. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 299.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 943.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 443. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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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
−8.747807040402326329020025411935, −8.106137529684470556688395679255, −7.45154164376702118690564519038, −6.22566005777597742026995183148, −5.67093780219743106740325979147, −4.53759942600252357630155214016, −3.57292140568406433679999687611, −2.03714013342576036549107376016, −1.09109862497464628654397917523, −0.10925737787235762421973151645,
2.08697124004289894218962419174, 3.12797331220385186908131125349, 3.88043137484788058246384999060, 4.81026649157314861280491380484, 6.15232530021591832749580873146, 6.81712803476495746275377785807, 7.38347780405096529477203116757, 8.795347286511747317533939541615, 9.633468796054178963694805071780, 10.07390204957860609751471696254