L(s) = 1 | + 9i·3-s + 89.8·5-s + 112.·7-s − 81·9-s + (346. + 202. i)11-s − 898. i·13-s + 808. i·15-s − 13.3i·17-s + 2.20e3·19-s + 1.00e3i·21-s − 2.28e3i·23-s + 4.94e3·25-s − 729i·27-s + 2.52e3i·29-s − 6.30e3i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.60·5-s + 0.864·7-s − 0.333·9-s + (0.863 + 0.503i)11-s − 1.47i·13-s + 0.927i·15-s − 0.0112i·17-s + 1.40·19-s + 0.499i·21-s − 0.901i·23-s + 1.58·25-s − 0.192i·27-s + 0.558i·29-s − 1.17i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00412i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.849954140\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.849954140\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
| 11 | \( 1 + (-346. - 202. i)T \) |
good | 5 | \( 1 - 89.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 112.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 898. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 13.3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.28e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.52e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.30e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.83e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.88e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 77.7T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.77e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.34e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.09e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.02e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.07e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.33e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 5.45e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 1.04e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.22e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.05e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.32e4T + 8.58e9T^{2} \) |
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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
−9.969290885707352638460216947504, −9.412813719993236594437152776721, −8.455096636631453476794833304175, −7.37189538835962893657427914194, −6.10790561222530240164457843325, −5.42035429107331072821306031087, −4.59931215612511559218662459535, −3.14584758366261228916484544300, −2.01388920125546773189664476875, −0.935745825674435742336405128899,
1.35165875818728881753918476780, 1.62626307494443968175966931095, 2.98830082362972246364196125266, 4.55967638313655161179603219970, 5.60407410681256526642461815684, 6.34694502774689602954566439626, 7.20097923679983388897318459866, 8.363271376624937417224622339005, 9.354462707302113700599254398357, 9.733707041262888183312355988931