| L(s) = 1 | − 0.792i·3-s − 1.37·5-s + 2.37·9-s − 3.31i·11-s + 1.08i·15-s − 6.13i·23-s − 3.11·25-s − 4.25i·27-s − 9.30i·31-s − 2.62·33-s + 12.1·37-s − 3.25·45-s − 6.63i·47-s − 7·49-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.457i·3-s − 0.613·5-s + 0.790·9-s − 1.00i·11-s + 0.280i·15-s − 1.27i·23-s − 0.623·25-s − 0.819i·27-s − 1.67i·31-s − 0.457·33-s + 1.99·37-s − 0.485·45-s − 0.967i·47-s − 49-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.887719 - 0.887719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.887719 - 0.887719i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + 3.31iT \) |
| good | 3 | \( 1 + 0.792iT - 3T^{2} \) |
| 5 | \( 1 + 1.37T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6.13iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 9.30iT - 31T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.63iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 14.6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 16.2iT - 67T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 18.8T + 89T^{2} \) |
| 97 | \( 1 + 0.116T + 97T^{2} \) |
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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
−10.22813794644181192661398772418, −9.389496241103099901677203854216, −8.210809527127793061603495254457, −7.78929933593152869980281154206, −6.69071688197310778115376908741, −5.93832061135932763866507331135, −4.57232690369064062137807117199, −3.71311839794002161989269748394, −2.33944499646651753526895157690, −0.69872148057116350916872204151,
1.60765128419964104500708531684, 3.25040194358592486475394653805, 4.27448998901315066900756194481, 4.95024248034381384755391670478, 6.27378809131363283125655169805, 7.37403041010297364130305834221, 7.84020643354933590453384925189, 9.185431394470184655989147048611, 9.743809543958980987785687291482, 10.59245793058763409246316934710
