L(s) = 1 | − 2.64i·3-s + 2.23·5-s − 2.64i·7-s − 4.00·9-s + 5.91i·11-s + 6.70·13-s − 5.91i·15-s + 2.23·17-s − 7.00·21-s + 5.00·25-s + 2.64i·27-s + 9·29-s + 15.6·33-s − 5.91i·35-s − 17.7i·39-s + ⋯ |
L(s) = 1 | − 1.52i·3-s + 0.999·5-s − 0.999i·7-s − 1.33·9-s + 1.78i·11-s + 1.86·13-s − 1.52i·15-s + 0.542·17-s − 1.52·21-s + 1.00·25-s + 0.509i·27-s + 1.67·29-s + 2.72·33-s − 0.999i·35-s − 2.84i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.471687406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.471687406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 2.64iT - 3T^{2} \) |
| 11 | \( 1 - 5.91iT - 11T^{2} \) |
| 13 | \( 1 - 6.70T + 13T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 7.93iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 17.7iT - 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−8.626306891505046435583516114331, −7.969079206824635799565574942351, −7.08003539853234198774804706796, −6.68013110300086591175341991323, −6.01998656101345551648567980017, −4.97022219418199792619987377017, −3.90622143149486664415281225481, −2.63373764384833067363313644719, −1.59798798006578019082320099172, −1.09084349034119661456841205307,
1.24398580184555648022699895802, 2.89924399571129366147212875394, 3.32410321267246777402050249021, 4.42838289832975287294250564108, 5.42659105280613253975064734232, 5.91471758231783086853878811575, 6.37699248845258382506900904825, 8.185581312362562563975621599331, 8.803905465785640655266937253567, 9.039538471472936217651485159198