| L(s) = 1 | − 4.86·2-s + 15.6·4-s + 12.8·5-s + 25.7·7-s − 37.4·8-s − 62.7·10-s + 53.1·11-s − 81.7·13-s − 125.·14-s + 56.6·16-s − 36.6·17-s + 7.96·19-s + 202.·20-s − 258.·22-s + 53.4·23-s + 41.0·25-s + 397.·26-s + 403.·28-s − 104.·29-s + 18.8·31-s + 23.5·32-s + 178.·34-s + 331.·35-s − 258.·37-s − 38.7·38-s − 482.·40-s + 136.·41-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.96·4-s + 1.15·5-s + 1.38·7-s − 1.65·8-s − 1.98·10-s + 1.45·11-s − 1.74·13-s − 2.38·14-s + 0.885·16-s − 0.523·17-s + 0.0962·19-s + 2.26·20-s − 2.50·22-s + 0.484·23-s + 0.328·25-s + 3.00·26-s + 2.72·28-s − 0.666·29-s + 0.109·31-s + 0.130·32-s + 0.900·34-s + 1.59·35-s − 1.14·37-s − 0.165·38-s − 1.90·40-s + 0.521·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 4.86T + 8T^{2} \) |
| 5 | \( 1 - 12.8T + 125T^{2} \) |
| 7 | \( 1 - 25.7T + 343T^{2} \) |
| 11 | \( 1 - 53.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 81.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.96T + 6.85e3T^{2} \) |
| 23 | \( 1 - 53.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 18.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 136.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 430.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 11.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 706.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 552.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 986.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 144.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 507.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 502.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.29e3T + 9.12e5T^{2} \) |
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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
−8.617117810595044109481117120893, −7.61925803174330615387310112039, −7.09373312007621874444948691490, −6.30243441350036438204910235725, −5.26818166722541994158037384691, −4.42494863079499860708474869964, −2.72391981021564740491837767962, −1.72692538485903086058359414155, −1.48087284843943639052396584712, 0,
1.48087284843943639052396584712, 1.72692538485903086058359414155, 2.72391981021564740491837767962, 4.42494863079499860708474869964, 5.26818166722541994158037384691, 6.30243441350036438204910235725, 7.09373312007621874444948691490, 7.61925803174330615387310112039, 8.617117810595044109481117120893
