L(s) = 1 | − i·2-s − i·3-s − 4-s + (−1 − 2i)5-s − 6-s − i·7-s + i·8-s − 9-s + (−2 + i)10-s − 3·11-s + i·12-s − 4i·13-s − 14-s + (−2 + i)15-s + 16-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (−0.632 + 0.316i)10-s − 0.904·11-s + 0.288i·12-s − 1.10i·13-s − 0.267·14-s + (−0.516 + 0.258i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.305556 + 0.494401i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305556 + 0.494401i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 5iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 5iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 5iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.545010160273345635842361833700, −8.627607386761118794623801111332, −7.86416968838300278318250902741, −7.30237574232669532135802547708, −5.69711276393135988105786699428, −5.12899848982183335966401969341, −3.92168080914936600301426973713, −2.92225776139589165403907765123, −1.51866075989609014854314795593, −0.27446055720847916293724345868,
2.40531935874045117295226274603, 3.54288568469892368614829201748, 4.52910691332778202043452752015, 5.43887489988290248425580158710, 6.48148449414858039901327596220, 7.14030689535698297919481576656, 8.154227292792793092578435497507, 8.800218478491754257952755942119, 9.860805618875227593481910657206, 10.44008843758896986894137273194