L(s) = 1 | − 2.22i·2-s + 2.75i·3-s − 2.94·4-s + 6.11·6-s − 1.28i·7-s + 2.08i·8-s − 4.57·9-s + 0.693·11-s − 8.09i·12-s − 4.64i·13-s − 2.85·14-s − 1.23·16-s + 1.40i·17-s + 10.1i·18-s + 3.40·19-s + ⋯ |
L(s) = 1 | − 1.57i·2-s + 1.58i·3-s − 1.47·4-s + 2.49·6-s − 0.484i·7-s + 0.738i·8-s − 1.52·9-s + 0.209·11-s − 2.33i·12-s − 1.28i·13-s − 0.761·14-s − 0.308·16-s + 0.341i·17-s + 2.39i·18-s + 0.782·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.276957365\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.276957365\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 + iT \) |
good | 2 | \( 1 + 2.22iT - 2T^{2} \) |
| 3 | \( 1 - 2.75iT - 3T^{2} \) |
| 7 | \( 1 + 1.28iT - 7T^{2} \) |
| 11 | \( 1 - 0.693T + 11T^{2} \) |
| 13 | \( 1 + 4.64iT - 13T^{2} \) |
| 17 | \( 1 - 1.40iT - 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 3.45iT - 23T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 + 4.95T + 31T^{2} \) |
| 37 | \( 1 + 11.0iT - 37T^{2} \) |
| 41 | \( 1 - 7.25T + 41T^{2} \) |
| 47 | \( 1 + 13.1iT - 47T^{2} \) |
| 53 | \( 1 + 6.42iT - 53T^{2} \) |
| 59 | \( 1 + 0.115T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 9.51iT - 67T^{2} \) |
| 71 | \( 1 + 0.745T + 71T^{2} \) |
| 73 | \( 1 - 4.37iT - 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 - 8.69iT - 97T^{2} \) |
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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
−10.01916946144827929312101494137, −9.198690691515015145799995249487, −8.520669614634224929911047219002, −7.26151999265308915643189081728, −5.69488788578594772570773889920, −4.90293803811081926227004171792, −3.88057052501216918803268572022, −3.51507902037133581354864155169, −2.38987565249820019533658056038, −0.61796378754525042527185623509,
1.37531385273089020552565174252, 2.66978898195202485018462988431, 4.38553236032962417402832125327, 5.53244587353711156677731175889, 6.15350278473162142129572874618, 6.95840563893210253276861206454, 7.36802793197956628352469190456, 8.176461822399670594265168214258, 8.936202898839496793860618912725, 9.615469039332536772406486395156