L(s) = 1 | − 3.46i·5-s − i·7-s + 17.3·11-s − 1.73i·13-s − 6·17-s − 1.73·19-s − 30i·23-s + 13.0·25-s + 20.7i·29-s − 14i·31-s − 3.46·35-s − 19.0i·37-s + 48·41-s − 20.7·43-s + 66i·47-s + ⋯ |
L(s) = 1 | − 0.692i·5-s − 0.142i·7-s + 1.57·11-s − 0.133i·13-s − 0.352·17-s − 0.0911·19-s − 1.30i·23-s + 0.520·25-s + 0.716i·29-s − 0.451i·31-s − 0.0989·35-s − 0.514i·37-s + 1.17·41-s − 0.483·43-s + 1.40i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.092709251\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.092709251\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.46iT - 25T^{2} \) |
| 7 | \( 1 + iT - 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 + 1.73iT - 169T^{2} \) |
| 17 | \( 1 + 6T + 289T^{2} \) |
| 19 | \( 1 + 1.73T + 361T^{2} \) |
| 23 | \( 1 + 30iT - 529T^{2} \) |
| 29 | \( 1 - 20.7iT - 841T^{2} \) |
| 31 | \( 1 + 14iT - 961T^{2} \) |
| 37 | \( 1 + 19.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 48T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 66iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 31.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 43.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 48iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49T + 5.32e3T^{2} \) |
| 79 | \( 1 + 83iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 66T + 7.92e3T^{2} \) |
| 97 | \( 1 - 107T + 9.40e3T^{2} \) |
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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
−9.038692993567060308404804069770, −8.359836244888864156660992807203, −7.34362311790104023198774684173, −6.53602609728570712747593987695, −5.82677101645902832071840530885, −4.61599650135679255199978754908, −4.17290698651573591866608294349, −2.95932528268430346684067756955, −1.63739706059492798699868198898, −0.63726281208250494753165940225,
1.14710340769354465086855308218, 2.30427890343271921586680628333, 3.44954046010225112505127528949, 4.14123059712185716588148786618, 5.28121531069069437291103261866, 6.28816077400684475032079229916, 6.81443530191386726180363359263, 7.59467352107233719936577664775, 8.664863908975268124298947405575, 9.258952121333707599542839132186