L(s) = 1 | + 2.82i·3-s + i·5-s − 2.82·7-s − 5.00·9-s + 5.65i·11-s + 2i·13-s − 2.82·15-s + 2·17-s − 8.00i·21-s + 2.82·23-s − 25-s − 5.65i·27-s − 6i·29-s − 5.65·31-s − 16.0·33-s + ⋯ |
L(s) = 1 | + 1.63i·3-s + 0.447i·5-s − 1.06·7-s − 1.66·9-s + 1.70i·11-s + 0.554i·13-s − 0.730·15-s + 0.485·17-s − 1.74i·21-s + 0.589·23-s − 0.200·25-s − 1.08i·27-s − 1.11i·29-s − 1.01·31-s − 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8668849479\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8668849479\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 3 | \( 1 - 2.82iT - 3T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + 2.82iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.940096655610559686671560551760, −9.581726885645358553047146033818, −9.049696934652772412553276282570, −7.67661100397263353365656214357, −6.86137816883771863988117360982, −5.91061639709618980634821449642, −4.91337796002406823701912059557, −4.12360070308573337447723781820, −3.40064548928288007145308682572, −2.29428942092811512260598712903,
0.37258392630966781462872934426, 1.36067334862514455836791140180, 2.86425756751636377550836488887, 3.49110802942974164431421866743, 5.34084132463577479895481249631, 5.93614132116438903279411387559, 6.73319427278279329214541701374, 7.41782094965223287842466604287, 8.444296096422465025948149731556, 8.753819101325881482354829221044