| L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s + 2·7-s − i·8-s + 9-s − 10-s − 4·11-s + 12-s + 2i·13-s + 2i·14-s − i·15-s + 16-s − 2i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s + 0.755·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554i·13-s + 0.534i·14-s − 0.258i·15-s + 0.250·16-s − 0.485i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6272084179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6272084179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 37 | \( 1 + (6 - i)T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 10iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−10.25233601242330338090564003971, −9.540082499143076009899418847400, −8.275500262423017108059551941863, −7.82373939850816971140897808123, −6.90922549752020937169875704460, −6.08364303326375243717514241185, −5.19831875646926909265083324229, −4.55093287159720382558945460654, −3.23654449668267964420466352171, −1.70132295377990458461839531936,
0.29542925061326266326070552891, 1.72754324329536463598166408976, 2.91158535905479085598150828743, 4.24763845043794295734265975699, 5.10598899958593967280850762312, 5.58482274861421329117222928399, 6.97706187661899317846342614308, 7.941092806026474074300442167043, 8.606803320091890227767562289421, 9.549344045755351280133870684840
