| L(s) = 1 | − 3·5-s − 3·7-s + 2·11-s − 2·17-s − 3·19-s + 2·23-s + 4·25-s − 6·31-s + 9·35-s − 4·37-s − 10·41-s − 2·43-s − 3·47-s + 2·49-s − 3·53-s − 6·55-s − 12·59-s + 61-s + 10·67-s + 10·71-s + 7·73-s − 6·77-s − 8·79-s + 6·83-s + 6·85-s + 6·89-s + 9·95-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s + 0.603·11-s − 0.485·17-s − 0.688·19-s + 0.417·23-s + 4/5·25-s − 1.07·31-s + 1.52·35-s − 0.657·37-s − 1.56·41-s − 0.304·43-s − 0.437·47-s + 2/7·49-s − 0.412·53-s − 0.809·55-s − 1.56·59-s + 0.128·61-s + 1.22·67-s + 1.18·71-s + 0.819·73-s − 0.683·77-s − 0.900·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s + 0.923·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 463 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.25426426599751, −12.72501784145403, −12.33629640943926, −11.98446831876664, −11.44348147183864, −10.90458169038408, −10.72206884126722, −9.856924891428193, −9.597647903995791, −8.931010595004789, −8.624907597107458, −8.057028218675580, −7.571582424309158, −6.972565832172759, −6.552475462802546, −6.370138978538362, −5.414129015550212, −4.980312668147453, −4.279311035916459, −3.845596367424385, −3.393859474984159, −3.026381934612378, −2.129232694870973, −1.506735252762241, −0.5095291574319620, 0,
0.5095291574319620, 1.506735252762241, 2.129232694870973, 3.026381934612378, 3.393859474984159, 3.845596367424385, 4.279311035916459, 4.980312668147453, 5.414129015550212, 6.370138978538362, 6.552475462802546, 6.972565832172759, 7.571582424309158, 8.057028218675580, 8.624907597107458, 8.931010595004789, 9.597647903995791, 9.856924891428193, 10.72206884126722, 10.90458169038408, 11.44348147183864, 11.98446831876664, 12.33629640943926, 12.72501784145403, 13.25426426599751
