L(s) = 1 | + i·3-s + (−1 + 2i)5-s + i·7-s − 9-s + 2·11-s + 2i·13-s + (−2 − i)15-s − 2·19-s − 21-s + 8i·23-s + (−3 − 4i)25-s − i·27-s − 2·29-s − 6·31-s + 2i·33-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.447 + 0.894i)5-s + 0.377i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + (−0.516 − 0.258i)15-s − 0.458·19-s − 0.218·21-s + 1.66i·23-s + (−0.600 − 0.800i)25-s − 0.192i·27-s − 0.371·29-s − 1.07·31-s + 0.348i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.233023 + 0.987102i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.233023 + 0.987102i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 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
−10.67891358246852338753424523886, −9.611823436722503069938489258341, −9.099676340522459889639882847100, −7.988456332843410587155092074632, −7.12164845569631818335312751872, −6.24656679780223388853501204015, −5.27261076885941236528605010110, −4.01954853574863734770663621304, −3.37813778600513446651741199482, −2.01192264765215878367684374772,
0.48936536022524026085883536692, 1.82612035316375870837682414590, 3.39858133999925403934209529644, 4.43007500352404244440486744602, 5.37199928499264628790821013458, 6.49522545600183232242990988639, 7.26175853751102238744055598102, 8.335427401241195895460483196826, 8.669646181919359046844088089535, 9.822926892149583366672422936997