| L(s) = 1 | + 0.414i·2-s − i·3-s + 1.82·4-s + 0.414·6-s + 2.82i·7-s + 1.58i·8-s − 9-s − 2·11-s − 1.82i·12-s + i·13-s − 1.17·14-s + 3·16-s + 7.65i·17-s − 0.414i·18-s + 2.82·19-s + ⋯ |
| L(s) = 1 | + 0.292i·2-s − 0.577i·3-s + 0.914·4-s + 0.169·6-s + 1.06i·7-s + 0.560i·8-s − 0.333·9-s − 0.603·11-s − 0.527i·12-s + 0.277i·13-s − 0.313·14-s + 0.750·16-s + 1.85i·17-s − 0.0976i·18-s + 0.648·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55742 + 0.962544i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55742 + 0.962544i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 7.65iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 - 1.65iT - 43T^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6.82iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 0.343iT - 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.65iT - 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.65iT - 97T^{2} \) |
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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
−10.26682626831866304808148416165, −9.137620626934817216372194234813, −8.286628617779579779129729530131, −7.63134241965282597773165969978, −6.77064957575023757283151500966, −5.79333312753433223489266846666, −5.47253410739886357276672258338, −3.69591931518558975956657042016, −2.49655701525501391356363088685, −1.70230917544530293627244927122,
0.839522067862243753089189013516, 2.56134256279391386554744594220, 3.33549921202542501754483141298, 4.50108332377588950532099191872, 5.40731166354287391844085444498, 6.57806081979420055479487990172, 7.36256996510191464630543754572, 7.993056984177906952990905357027, 9.371065066976494954544880111367, 9.980241476775738039752219111799
