| L(s) = 1 | − i·2-s − 4-s + (1.89 + 1.19i)5-s − 1.91i·7-s + i·8-s + (1.19 − 1.89i)10-s − 6.34·11-s + 1.16i·13-s − 1.91·14-s + 16-s + 0.561i·17-s − 0.439·19-s + (−1.89 − 1.19i)20-s + 6.34i·22-s + 1.39i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.846 + 0.532i)5-s − 0.722i·7-s + 0.353i·8-s + (0.376 − 0.598i)10-s − 1.91·11-s + 0.321i·13-s − 0.510·14-s + 0.250·16-s + 0.136i·17-s − 0.100·19-s + (−0.423 − 0.266i)20-s + 1.35i·22-s + 0.290i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758165051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758165051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-1.89 - 1.19i)T \) |
| 37 | \( 1 - iT \) |
| good | 7 | \( 1 + 1.91iT - 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 - 1.16iT - 13T^{2} \) |
| 17 | \( 1 - 0.561iT - 17T^{2} \) |
| 19 | \( 1 + 0.439T + 19T^{2} \) |
| 23 | \( 1 - 1.39iT - 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 - 6.09T + 31T^{2} \) |
| 41 | \( 1 - 4.97T + 41T^{2} \) |
| 43 | \( 1 + 4.94iT - 43T^{2} \) |
| 47 | \( 1 - 6.69iT - 47T^{2} \) |
| 53 | \( 1 + 9.13iT - 53T^{2} \) |
| 59 | \( 1 - 5.51T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 5.03iT - 67T^{2} \) |
| 71 | \( 1 - 3.59T + 71T^{2} \) |
| 73 | \( 1 + 5.07iT - 73T^{2} \) |
| 79 | \( 1 - 0.343T + 79T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−8.562316526382497344485783769038, −7.903500163974607187004118183823, −7.10034567675261964957604431165, −6.28122375409222548576773191101, −5.36531312590972763328713449432, −4.75633691049674172368728298376, −3.69211774233325381548485602226, −2.73064212206343883354935777407, −2.18006580516240574958033037320, −0.834237899254373491560863718884,
0.73069690479785477573737672906, 2.31126235725857293059668935015, 2.87201569416183250824887847887, 4.41648227486604416578023590106, 5.12332818207896035564849404709, 5.62994024514398680684593918575, 6.27099670754606399081960069535, 7.22533707519945570713267997263, 8.143893536639284464748674904272, 8.482340297128131000183058069705
