| L(s) = 1 | + i·2-s − i·3-s + 4-s + 6-s + 3i·8-s − 9-s − 4·11-s − i·12-s − 2i·13-s − 16-s − 2i·17-s − i·18-s − 4·19-s − 4i·22-s + 3·24-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s + 0.5·4-s + 0.408·6-s + 1.06i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s − 0.554i·13-s − 0.250·16-s − 0.485i·17-s − 0.235i·18-s − 0.917·19-s − 0.852i·22-s + 0.612·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.982545 + 0.231947i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.982545 + 0.231947i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−14.82600110360146444091838197656, −13.60501366116669137684821098129, −12.59166450815821389290892738729, −11.37619323081471910438366922608, −10.30336279752583334125713240413, −8.436983745411476596171759227745, −7.59119348029160234777087323207, −6.43506067565300041466428256615, −5.21819757671916085151264765739, −2.59227805750375734226524276590,
2.50577656933738056614132106187, 4.17314498147057539715128367537, 5.95632579760025348206124952315, 7.52156633457050854866829125129, 9.066268324289895262971809662894, 10.40433512253293882895387196372, 10.91103772036436981090062822670, 12.19251527247341498052026940081, 13.12990429596880409303772465604, 14.56809004589719542184363179136
