| L(s) = 1 | − 352.·2-s − 6.56e3·3-s − 6.49e3·4-s + 2.31e6·6-s − 1.07e7·7-s + 4.85e7·8-s + 4.30e7·9-s + 1.00e9·11-s + 4.25e7·12-s − 3.43e9·13-s + 3.77e9·14-s − 1.62e10·16-s − 4.81e10·17-s − 1.51e10·18-s + 8.08e9·19-s + 7.02e10·21-s − 3.54e11·22-s + 3.08e11·23-s − 3.18e11·24-s + 1.21e12·26-s − 2.82e11·27-s + 6.94e10·28-s + 7.97e11·29-s − 3.17e12·31-s − 6.15e11·32-s − 6.58e12·33-s + 1.69e13·34-s + ⋯ |
| L(s) = 1 | − 0.974·2-s − 0.577·3-s − 0.0495·4-s + 0.562·6-s − 0.701·7-s + 1.02·8-s + 0.333·9-s + 1.41·11-s + 0.0285·12-s − 1.16·13-s + 0.683·14-s − 0.948·16-s − 1.67·17-s − 0.324·18-s + 0.109·19-s + 0.405·21-s − 1.37·22-s + 0.820·23-s − 0.590·24-s + 1.13·26-s − 0.192·27-s + 0.0347·28-s + 0.296·29-s − 0.668·31-s − 0.0989·32-s − 0.815·33-s + 1.63·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 352.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 1.07e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.00e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.43e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 4.81e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 8.08e9T + 5.48e21T^{2} \) |
| 23 | \( 1 - 3.08e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 7.97e11T + 7.25e24T^{2} \) |
| 31 | \( 1 + 3.17e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.61e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 7.63e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.29e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.44e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 8.60e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.21e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 4.27e14T + 2.24e30T^{2} \) |
| 67 | \( 1 - 1.16e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 3.01e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 7.32e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 7.07e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 1.18e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 5.69e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 8.11e16T + 5.95e33T^{2} \) |
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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
−10.47223710517683609081144269440, −9.431131941371853998247364746720, −8.855128161346867013976142612340, −7.24934990968072451796307912546, −6.55328219690790516150394447425, −4.96212999564976014538957491104, −3.92276642971808452743018285612, −2.15870462980458677366354752332, −0.888640997163160059385205579590, 0,
0.888640997163160059385205579590, 2.15870462980458677366354752332, 3.92276642971808452743018285612, 4.96212999564976014538957491104, 6.55328219690790516150394447425, 7.24934990968072451796307912546, 8.855128161346867013976142612340, 9.431131941371853998247364746720, 10.47223710517683609081144269440
