| L(s) = 1 | + 256·2-s + 6.55e4·4-s − 3.90e5·5-s + 2.54e7·7-s + 1.67e7·8-s − 1.00e8·10-s + 2.81e8·11-s − 1.52e9·13-s + 6.52e9·14-s + 4.29e9·16-s − 5.46e10·17-s + 6.88e8·19-s − 2.56e10·20-s + 7.19e10·22-s − 3.91e11·23-s + 1.52e11·25-s − 3.90e11·26-s + 1.66e12·28-s − 5.12e12·29-s − 7.31e10·31-s + 1.09e12·32-s − 1.40e13·34-s − 9.94e12·35-s − 6.81e12·37-s + 1.76e11·38-s − 6.55e12·40-s + 5.76e13·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.66·7-s + 0.353·8-s − 0.316·10-s + 0.395·11-s − 0.518·13-s + 1.18·14-s + 0.250·16-s − 1.90·17-s + 0.00930·19-s − 0.223·20-s + 0.279·22-s − 1.04·23-s + 0.200·25-s − 0.366·26-s + 0.834·28-s − 1.90·29-s − 0.0153·31-s + 0.176·32-s − 1.34·34-s − 0.746·35-s − 0.319·37-s + 0.00658·38-s − 0.158·40-s + 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 256T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 3.90e5T \) |
| good | 7 | \( 1 - 2.54e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 2.81e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.52e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 5.46e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 6.88e8T + 5.48e21T^{2} \) |
| 23 | \( 1 + 3.91e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 5.12e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 7.31e10T + 2.25e25T^{2} \) |
| 37 | \( 1 + 6.81e12T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.76e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.57e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 4.60e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.58e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 2.98e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 8.50e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 6.12e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 5.41e14T + 2.96e31T^{2} \) |
| 73 | \( 1 + 7.16e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 5.45e15T + 1.81e32T^{2} \) |
| 83 | \( 1 - 3.64e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 6.81e14T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.20e17T + 5.95e33T^{2} \) |
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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
−10.87240597225341458370583411905, −9.151402147488928727143610649744, −8.006317745274464613336810654567, −7.14655732106693150956963974270, −5.77288038162781346548127874775, −4.60557347054621249984079715936, −4.04492778674989145131922890956, −2.36986098489442391095351470652, −1.56482642609729583024643234745, 0,
1.56482642609729583024643234745, 2.36986098489442391095351470652, 4.04492778674989145131922890956, 4.60557347054621249984079715936, 5.77288038162781346548127874775, 7.14655732106693150956963974270, 8.006317745274464613336810654567, 9.151402147488928727143610649744, 10.87240597225341458370583411905
