| L(s) = 1 | + 0.732·2-s − 1.46·4-s − 3.86i·5-s − 4.38i·7-s − 2.53·8-s − 2.82i·10-s + 2.44i·13-s − 3.20i·14-s + 1.07·16-s − 3.46·17-s + 0.896i·19-s + 5.65i·20-s + 4.24i·23-s − 9.92·25-s + 1.79i·26-s + ⋯ |
| L(s) = 1 | + 0.517·2-s − 0.732·4-s − 1.72i·5-s − 1.65i·7-s − 0.896·8-s − 0.894i·10-s + 0.679i·13-s − 0.857i·14-s + 0.267·16-s − 0.840·17-s + 0.205i·19-s + 1.26i·20-s + 0.884i·23-s − 1.98·25-s + 0.351i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9326955830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9326955830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.732T + 2T^{2} \) |
| 5 | \( 1 + 3.86iT - 5T^{2} \) |
| 7 | \( 1 + 4.38iT - 7T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 0.896iT - 19T^{2} \) |
| 23 | \( 1 - 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 2.73T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 - 3.73T + 37T^{2} \) |
| 41 | \( 1 + 2.19T + 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 + 6.31iT - 47T^{2} \) |
| 53 | \( 1 + 8.38iT - 53T^{2} \) |
| 59 | \( 1 + 8.86iT - 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 1.19T + 67T^{2} \) |
| 71 | \( 1 + 8.76iT - 71T^{2} \) |
| 73 | \( 1 + 5.03iT - 73T^{2} \) |
| 79 | \( 1 - 5.79iT - 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 18.2iT - 89T^{2} \) |
| 97 | \( 1 + 7.53T + 97T^{2} \) |
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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
−9.460815810737230246904411601948, −8.644417818502469251921036296405, −7.959840065295171082710715190325, −6.91640644071799403432208224159, −5.76880046904930439047080089569, −4.78063941161779800669111880346, −4.34501893011440195652884527751, −3.61825974582625382618737559207, −1.51560713840839301525470472294, −0.36095503284257890623002008479,
2.49646597532610248424389764603, 2.91859571619178403073735613636, 4.09715820804017197080351248140, 5.26632154045005837401244074699, 6.03455344245560014102902830137, 6.64233513006178215115024219096, 7.85730835808139301665910239474, 8.710435917434522851052081042217, 9.438514529906828056761496358607, 10.31730652759846155646632880640
