| L(s) = 1 | − 1.63·3-s − 0.530·5-s − 0.325·7-s − 0.324·9-s + 3.05·11-s + 1.58·13-s + 0.867·15-s − 2.75·17-s − 1.04·19-s + 0.532·21-s + 2.36·23-s − 4.71·25-s + 5.43·27-s − 6.32·29-s + 7.66·31-s − 4.99·33-s + 0.172·35-s − 0.416·37-s − 2.59·39-s + 5.37·41-s − 9.93·43-s + 0.171·45-s − 3.44·47-s − 6.89·49-s + 4.50·51-s − 1.87·53-s − 1.62·55-s + ⋯ |
| L(s) = 1 | − 0.944·3-s − 0.237·5-s − 0.122·7-s − 0.108·9-s + 0.921·11-s + 0.439·13-s + 0.223·15-s − 0.667·17-s − 0.238·19-s + 0.116·21-s + 0.493·23-s − 0.943·25-s + 1.04·27-s − 1.17·29-s + 1.37·31-s − 0.870·33-s + 0.0291·35-s − 0.0684·37-s − 0.415·39-s + 0.840·41-s − 1.51·43-s + 0.0256·45-s − 0.502·47-s − 0.984·49-s + 0.630·51-s − 0.258·53-s − 0.218·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 1.63T + 3T^{2} \) |
| 5 | \( 1 + 0.530T + 5T^{2} \) |
| 7 | \( 1 + 0.325T + 7T^{2} \) |
| 11 | \( 1 - 3.05T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 - 2.36T + 23T^{2} \) |
| 29 | \( 1 + 6.32T + 29T^{2} \) |
| 31 | \( 1 - 7.66T + 31T^{2} \) |
| 37 | \( 1 + 0.416T + 37T^{2} \) |
| 41 | \( 1 - 5.37T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 16.7T + 71T^{2} \) |
| 73 | \( 1 + 7.62T + 73T^{2} \) |
| 79 | \( 1 - 8.44T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 - 4.93T + 97T^{2} \) |
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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
−9.212089713131262871797437536997, −8.480603334156073742669720815955, −7.50116735203553714052932559819, −6.41653033863665679298700953301, −6.13199377630027669879566295501, −4.98790805437922535027052151869, −4.16655740409495167825534522499, −3.07699200779516648207722651430, −1.52635882268899645968087467614, 0,
1.52635882268899645968087467614, 3.07699200779516648207722651430, 4.16655740409495167825534522499, 4.98790805437922535027052151869, 6.13199377630027669879566295501, 6.41653033863665679298700953301, 7.50116735203553714052932559819, 8.480603334156073742669720815955, 9.212089713131262871797437536997
