| L(s) = 1 | − 1.73i·3-s + 2.23·5-s − 2.99·9-s − 3.87i·15-s + 4.47·17-s − 7.74i·19-s − 3.46i·23-s + 5.00·25-s + 5.19i·27-s − 7.74i·31-s − 6.70·45-s + 10.3i·47-s − 7·49-s − 7.74i·51-s + 4.47·53-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 0.999·5-s − 0.999·9-s − 1.00i·15-s + 1.08·17-s − 1.77i·19-s − 0.722i·23-s + 1.00·25-s + 0.999i·27-s − 1.39i·31-s − 0.999·45-s + 1.51i·47-s − 49-s − 1.08i·51-s + 0.614·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27082 - 1.27082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27082 - 1.27082i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 7.74iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 7.74iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 7.74iT - 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.676492482428570393884584601817, −9.019560755612915415336738821995, −8.080206531722436283677385988856, −7.20355594826061677750358567218, −6.39965115768877804640050123198, −5.69111914169278744867702476633, −4.72242098198396998535138288387, −3.03452562343046857166825958169, −2.19816412844932628598084033043, −0.887647099559945148220158467806,
1.60744621765691428180051104265, 3.04533252504498375365702213481, 3.89724626102493859241685495299, 5.24706762779359754680770388262, 5.61729414620926474375690093995, 6.65111815742844143176614948544, 7.930986738657645955108358520206, 8.720070645406014723838013421605, 9.663779731819992042373158829490, 10.10068158264822088059502813555
