L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.707 + 0.707i)7-s − 0.999i·8-s + (−0.866 − 0.499i)10-s + (−1.67 − 0.965i)11-s − 0.517i·13-s + (−0.258 + 0.965i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)19-s − 0.999·20-s − 1.93·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.258 − 0.448i)26-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.707 + 0.707i)7-s − 0.999i·8-s + (−0.866 − 0.499i)10-s + (−1.67 − 0.965i)11-s − 0.517i·13-s + (−0.258 + 0.965i)14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)19-s − 0.999·20-s − 1.93·22-s + (−1.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.258 − 0.448i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.145689086\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145689086\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + 0.517iT - T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 1.93T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.775807083885434732517019624918, −7.980277731160168562890954758112, −7.26126061773492373011920079868, −5.89333081733351971476006700494, −5.60224114381731255958031963214, −4.91865923580093282451489948242, −3.75488609715582401164134277166, −3.09239839479635289213813817383, −2.17294522871916477944509236855, −0.51985678968160974480724341704,
2.29810375810028392088045361232, 2.98729726092772449629972594051, 3.98255777614693041762867323056, 4.53985793073211892777126467603, 5.64846101018920534644902016346, 6.39750828536180288546206028968, 7.17208859877260208620517576167, 7.65926937290322058065728319818, 8.197892742410080748151123097376, 9.616491675679873795603302739349