L(s) = 1 | + i·3-s − 9-s + 4·11-s − 2i·13-s + 2i·17-s − 8·19-s − 4i·23-s − i·27-s − 6·29-s + 4i·33-s + 2i·37-s + 2·39-s − 6·41-s + 4i·43-s − 12i·47-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.333·9-s + 1.20·11-s − 0.554i·13-s + 0.485i·17-s − 1.83·19-s − 0.834i·23-s − 0.192i·27-s − 1.11·29-s + 0.696i·33-s + 0.328i·37-s + 0.320·39-s − 0.937·41-s + 0.609i·43-s − 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−8.148999417763595037242873834441, −7.20954408718917121186469512149, −6.36133176608047715519177047249, −5.93574274819714052762172486210, −4.87662642103056694520551797063, −4.16413160757843991694599718276, −3.59876710541155682093588445546, −2.51633575047117498215008641202, −1.49785612267003571358929907306, 0,
1.45990141441098031357572591635, 2.10101473529025980216811474636, 3.29011294053351677908585571686, 4.09788697308073685171324925749, 4.83210746621997969870832160177, 5.98817181650402084242120908660, 6.35615423258339646253996866730, 7.17783413321688312870870099919, 7.71032510569072262086488957415