L(s) = 1 | − 2i·2-s − 4·4-s + 29i·7-s + 8i·8-s + 57·11-s − 20i·13-s + 58·14-s + 16·16-s + 72i·17-s + 106·19-s − 114i·22-s + 174i·23-s − 40·26-s − 116i·28-s − 210·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.56i·7-s + 0.353i·8-s + 1.56·11-s − 0.426i·13-s + 1.10·14-s + 0.250·16-s + 1.02i·17-s + 1.27·19-s − 1.10i·22-s + 1.57i·23-s − 0.301·26-s − 0.782i·28-s − 1.34·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.913945824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.913945824\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 29iT - 343T^{2} \) |
| 11 | \( 1 - 57T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 72iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 106T + 6.85e3T^{2} \) |
| 23 | \( 1 - 174iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 210T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 218iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 474iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 81iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 84T + 2.05e5T^{2} \) |
| 61 | \( 1 - 56T + 2.26e5T^{2} \) |
| 67 | \( 1 + 142iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 360T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 160T + 4.93e5T^{2} \) |
| 83 | \( 1 - 735iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 954T + 7.04e5T^{2} \) |
| 97 | \( 1 - 191iT - 9.12e5T^{2} \) |
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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
−9.348488303049488144379054139012, −8.859655166998024647620826184323, −7.996650752885588448376518018471, −6.89542905473033720653348949008, −5.71620871702143470721776051255, −5.41247446872237432445056676571, −3.94062863475010034017582896462, −3.29276604729768912207671742924, −2.07737396755249181123910991963, −1.24973816061674014263088229712,
0.49037539081063761991163497712, 1.38370381395521553047802982760, 3.19727429921465651927532247640, 4.16816466372746517723579701517, 4.68949183184355313443627819426, 5.97878596534447707558817415092, 6.85635989287206300495576340343, 7.23924360213739128865384479912, 8.101128957763037180460031134665, 9.270509076169233883855258397314