| L(s) = 1 | + 0.414·2-s + 3-s − 1.82·4-s − 5-s + 0.414·6-s − 3.41·7-s − 1.58·8-s + 9-s − 0.414·10-s + 2.82·11-s − 1.82·12-s − 5.41·13-s − 1.41·14-s − 15-s + 3·16-s − 4·17-s + 0.414·18-s − 2.82·19-s + 1.82·20-s − 3.41·21-s + 1.17·22-s − 6·23-s − 1.58·24-s + 25-s − 2.24·26-s + 27-s + 6.24·28-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.577·3-s − 0.914·4-s − 0.447·5-s + 0.169·6-s − 1.29·7-s − 0.560·8-s + 0.333·9-s − 0.130·10-s + 0.852·11-s − 0.527·12-s − 1.50·13-s − 0.377·14-s − 0.258·15-s + 0.750·16-s − 0.970·17-s + 0.0976·18-s − 0.648·19-s + 0.408·20-s − 0.745·21-s + 0.249·22-s − 1.25·23-s − 0.323·24-s + 0.200·25-s − 0.439·26-s + 0.192·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 + 15.8T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−10.32235555061267466180822521169, −9.444181284140794991678704857799, −9.077820780077449798814052905039, −7.909791602786781256767800212665, −6.90386867118158446352776102535, −5.87664044298959329900745955648, −4.40631948621539143839861037043, −3.83039348159519505383301947087, −2.55413117709499278887580036066, 0,
2.55413117709499278887580036066, 3.83039348159519505383301947087, 4.40631948621539143839861037043, 5.87664044298959329900745955648, 6.90386867118158446352776102535, 7.909791602786781256767800212665, 9.077820780077449798814052905039, 9.444181284140794991678704857799, 10.32235555061267466180822521169
