| L(s) = 1 | − 1.41i·2-s + (−2.88 + 0.831i)3-s − 2.00·4-s − 4.87i·5-s + (1.17 + 4.07i)6-s + 5.25·7-s + 2.82i·8-s + (7.61 − 4.79i)9-s − 6.89·10-s − 1.59i·11-s + (5.76 − 1.66i)12-s + 11.6·13-s − 7.43i·14-s + (4.05 + 14.0i)15-s + 4.00·16-s − 10.9i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.960 + 0.277i)3-s − 0.500·4-s − 0.975i·5-s + (0.195 + 0.679i)6-s + 0.750·7-s + 0.353i·8-s + (0.846 − 0.532i)9-s − 0.689·10-s − 0.144i·11-s + (0.480 − 0.138i)12-s + 0.895·13-s − 0.530i·14-s + (0.270 + 0.937i)15-s + 0.250·16-s − 0.642i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.119623 - 0.846322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.119623 - 0.846322i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (2.88 - 0.831i)T \) |
| 59 | \( 1 + 7.68iT \) |
| good | 5 | \( 1 + 4.87iT - 25T^{2} \) |
| 7 | \( 1 - 5.25T + 49T^{2} \) |
| 11 | \( 1 + 1.59iT - 121T^{2} \) |
| 13 | \( 1 - 11.6T + 169T^{2} \) |
| 17 | \( 1 + 10.9iT - 289T^{2} \) |
| 19 | \( 1 + 36.5T + 361T^{2} \) |
| 23 | \( 1 + 24.9iT - 529T^{2} \) |
| 29 | \( 1 - 1.87iT - 841T^{2} \) |
| 31 | \( 1 + 35.0T + 961T^{2} \) |
| 37 | \( 1 + 31.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 56.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 32.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 53.9iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 81.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 47.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 72.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.24T + 6.24e3T^{2} \) |
| 83 | \( 1 - 75.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 161. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 15.3T + 9.40e3T^{2} \) |
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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
−10.90409175826975723702985809438, −10.25942165007382630878022465964, −8.902424426708443179976631252684, −8.469273461082602618996763935763, −6.84332073579172409780818892136, −5.59113667288077903932137701264, −4.75388427456537815267838997924, −3.91989428315865882628642990734, −1.80907615240390647059029193638, −0.45596121311424479119078052214,
1.72064720558581772543471696269, 3.80833698233196194038320356922, 4.98921621117264194774647573453, 6.11848513979585235130877067136, 6.69764611764245830701305405675, 7.71160019476181338578665284872, 8.604172173014238005329712841729, 10.05504870089004171665030795852, 10.92167849778625735261739903149, 11.36752629013160904187800356229
