L(s) = 1 | + 2.82i·3-s + 2.82i·7-s − 5.00·9-s − 5.65·11-s − 2i·13-s − 2i·17-s − 8.00·21-s + 2.82i·23-s − 5.65i·27-s − 6·29-s + 5.65·31-s − 16.0i·33-s + 10i·37-s + 5.65·39-s + 2·41-s + ⋯ |
L(s) = 1 | + 1.63i·3-s + 1.06i·7-s − 1.66·9-s − 1.70·11-s − 0.554i·13-s − 0.485i·17-s − 1.74·21-s + 0.589i·23-s − 1.08i·27-s − 1.11·29-s + 1.01·31-s − 2.78i·33-s + 1.64i·37-s + 0.905·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.178142 - 0.754623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.178142 - 0.754623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.82iT - 3T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 2.82iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 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
−10.56024973642593481551583740055, −9.963796753873564761924285769464, −9.210697449861124544720869177237, −8.422988010222493273615062986096, −7.57269581173972281806426407098, −5.91812093958014625017984981495, −5.30050357466312350365644809524, −4.64426665543346457536299647370, −3.29222843418182693735341644634, −2.56254252655487383492855895410,
0.36415507574742291697408329294, 1.77580982020953030801139418006, 2.83476634108581820637337647699, 4.29675719743829133261626700486, 5.56444806224261619030748421578, 6.47459775121294863540613204168, 7.35449912654377963711685213329, 7.77356866603823740755352696653, 8.585209949607431252987563390872, 9.890432441547226913669602327236