| L(s) = 1 | + 5.65·2-s − 15.5·3-s + 32.0·4-s + 124. i·5-s − 88.1·6-s − 215. i·7-s + 181.·8-s + 243·9-s + 703. i·10-s + 961. i·11-s − 498.·12-s − 657.·13-s − 1.22e3i·14-s − 1.93e3i·15-s + 1.02e3·16-s − 1.21e3i·17-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.994i·5-s − 0.408·6-s − 0.629i·7-s + 0.353·8-s + 0.333·9-s + 0.703i·10-s + 0.722i·11-s − 0.288·12-s − 0.299·13-s − 0.444i·14-s − 0.574i·15-s + 0.250·16-s − 0.246i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.295221328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.295221328\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 5.65T \) |
| 3 | \( 1 + 15.5T \) |
| 23 | \( 1 + (9.70e3 + 7.33e3i)T \) |
| good | 5 | \( 1 - 124. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 215. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 961. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 657.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 1.21e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 9.15e3iT - 4.70e7T^{2} \) |
| 29 | \( 1 + 4.67e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.30e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 6.96e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.55e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 9.48e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.96e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 1.91e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.40e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.30e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.78e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.70e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 2.90e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.30e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 8.08e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 1.98e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 3.51e5iT - 8.32e11T^{2} \) |
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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
−12.42711621681065634831512052732, −11.49144188111173400251171992581, −10.55087039924110515781678229949, −9.825833579154859508588905433232, −7.75048949213824869544253725676, −6.91955232688426759353041030979, −5.92563970047314356665551501285, −4.56597832231823541321152797520, −3.39026192657210301952675017292, −1.79383255497502145831334762370,
0.31426280182497707494446612322, 1.96060900135330742843665987958, 3.74181715811728030578431427699, 5.12788871781923566942002943683, 5.70395444011466386936378825568, 7.10340911546225907234259291286, 8.517133314933092758067523663106, 9.528326995820624334250868994187, 11.03602738323150097023490999039, 11.73988512559265360516782165932
