| L(s) = 1 | + 1.71e5·2-s + 2.07e10·4-s + 2.01e11·5-s + 5.50e13·7-s + 2.08e15·8-s + 3.45e16·10-s + 8.18e16·11-s − 1.90e18·13-s + 9.42e18·14-s + 1.79e20·16-s + 3.33e20·17-s − 1.40e20·19-s + 4.18e21·20-s + 1.40e22·22-s − 3.12e22·23-s − 7.58e22·25-s − 3.26e23·26-s + 1.14e24·28-s + 1.50e24·29-s + 5.18e23·31-s + 1.27e25·32-s + 5.72e25·34-s + 1.10e25·35-s − 3.01e25·37-s − 2.40e25·38-s + 4.20e26·40-s + 2.18e26·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.41·4-s + 0.590·5-s + 0.625·7-s + 2.62·8-s + 1.09·10-s + 0.537·11-s − 0.793·13-s + 1.15·14-s + 2.43·16-s + 1.66·17-s − 0.111·19-s + 1.42·20-s + 0.993·22-s − 1.06·23-s − 0.651·25-s − 1.46·26-s + 1.51·28-s + 1.12·29-s + 0.128·31-s + 1.87·32-s + 3.07·34-s + 0.369·35-s − 0.401·37-s − 0.206·38-s + 1.54·40-s + 0.536·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(8.906251616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.906251616\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.71e5T + 8.58e9T^{2} \) |
| 5 | \( 1 - 2.01e11T + 1.16e23T^{2} \) |
| 7 | \( 1 - 5.50e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 8.18e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.90e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 3.33e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.40e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.12e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.50e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 5.18e23T + 1.64e49T^{2} \) |
| 37 | \( 1 + 3.01e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 2.18e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.76e27T + 8.02e53T^{2} \) |
| 47 | \( 1 + 3.25e27T + 1.51e55T^{2} \) |
| 53 | \( 1 + 9.17e27T + 7.96e56T^{2} \) |
| 59 | \( 1 - 1.18e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 9.92e27T + 8.23e58T^{2} \) |
| 67 | \( 1 - 1.11e30T + 1.82e60T^{2} \) |
| 71 | \( 1 + 7.58e29T + 1.23e61T^{2} \) |
| 73 | \( 1 + 6.06e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 5.57e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 4.13e31T + 2.13e63T^{2} \) |
| 89 | \( 1 + 6.21e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 4.04e32T + 3.65e65T^{2} \) |
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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
−14.08015330685980784913447413397, −12.54422226576142171013309887400, −11.65882201042686300120446471056, −10.07289973263252693515985927038, −7.66309016926825645684712359703, −6.18307145153677745740702837569, −5.19962423923673568072910494536, −4.00795900289828917689815351764, −2.62532772055263355972238419670, −1.46123154365470444616523474221,
1.46123154365470444616523474221, 2.62532772055263355972238419670, 4.00795900289828917689815351764, 5.19962423923673568072910494536, 6.18307145153677745740702837569, 7.66309016926825645684712359703, 10.07289973263252693515985927038, 11.65882201042686300120446471056, 12.54422226576142171013309887400, 14.08015330685980784913447413397
