| L(s) = 1 | − 74.3i·3-s − 154.·5-s − 1.58e3·7-s + 1.03e3·9-s + 2.55e4·11-s − 1.03e4i·13-s + 1.15e4i·15-s − 1.32e5·17-s + (−1.31e4 + 1.29e5i)19-s + 1.17e5i·21-s + 3.34e5·23-s − 3.66e5·25-s − 5.64e5i·27-s + 6.05e5i·29-s + 1.41e5i·31-s + ⋯ |
| L(s) = 1 | − 0.917i·3-s − 0.247·5-s − 0.660·7-s + 0.158·9-s + 1.74·11-s − 0.363i·13-s + 0.227i·15-s − 1.59·17-s + (−0.100 + 0.994i)19-s + 0.605i·21-s + 1.19·23-s − 0.938·25-s − 1.06i·27-s + 0.855i·29-s + 0.153i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.7081841012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7081841012\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.31e4 - 1.29e5i)T \) |
| good | 3 | \( 1 + 74.3iT - 6.56e3T^{2} \) |
| 5 | \( 1 + 154.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 1.58e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 2.55e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 1.03e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 1.32e5T + 6.97e9T^{2} \) |
| 23 | \( 1 - 3.34e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 6.05e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.41e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 2.45e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.38e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 1.53e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 5.01e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 1.18e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 7.32e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.12e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.87e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 2.17e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 8.89e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 5.26e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 3.76e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.06e8iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 2.45e7iT - 7.83e15T^{2} \) |
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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
−10.62890842411380971519614712100, −9.414422693221479612035176281515, −8.715299145572350039536256346934, −7.43841724708829371406590353402, −6.73932919944377782329947270690, −6.03057036996078456073834652600, −4.38830134431417899251001779507, −3.44731964726055547894329330116, −1.97759930201488409712007575023, −1.07401808090158163749772850879,
0.15453033699238056700562562657, 1.61876876556921486232482885646, 3.15689100081598301522169153543, 4.14621355719319365259950506228, 4.75699192465680217940058544775, 6.45867365351570242802286018722, 6.87913050693126838962670178806, 8.527387181297712346219845690844, 9.390658383014857467993172086853, 9.806489946291174710857595919927
