| L(s) = 1 | + 3-s + 1.82·5-s − 2.26·7-s + 9-s + 0.576·11-s − 1.28·13-s + 1.82·15-s − 1.88·17-s − 5.66·19-s − 2.26·21-s + 2.15·23-s − 1.66·25-s + 27-s + 6.70·29-s − 2.07·31-s + 0.576·33-s − 4.14·35-s − 5.63·37-s − 1.28·39-s + 8.23·41-s − 2.54·43-s + 1.82·45-s − 9.37·47-s − 1.84·49-s − 1.88·51-s + 6.17·53-s + 1.05·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.816·5-s − 0.857·7-s + 0.333·9-s + 0.173·11-s − 0.355·13-s + 0.471·15-s − 0.456·17-s − 1.29·19-s − 0.495·21-s + 0.449·23-s − 0.332·25-s + 0.192·27-s + 1.24·29-s − 0.372·31-s + 0.100·33-s − 0.700·35-s − 0.926·37-s − 0.205·39-s + 1.28·41-s − 0.388·43-s + 0.272·45-s − 1.36·47-s − 0.264·49-s − 0.263·51-s + 0.848·53-s + 0.141·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.82T + 5T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 11 | \( 1 - 0.576T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 1.88T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 6.70T + 29T^{2} \) |
| 31 | \( 1 + 2.07T + 31T^{2} \) |
| 37 | \( 1 + 5.63T + 37T^{2} \) |
| 41 | \( 1 - 8.23T + 41T^{2} \) |
| 43 | \( 1 + 2.54T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 - 3.25T + 59T^{2} \) |
| 61 | \( 1 - 1.95T + 61T^{2} \) |
| 67 | \( 1 + 8.37T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 3.01T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−7.39824793347898235416004593275, −6.68460279126830487651733874171, −6.29493150513934428120131310272, −5.46006811073047706869999706549, −4.58104496370043598564475955285, −3.86923035092990271475285611739, −2.93727634776662109083862804418, −2.34503180134762693604375030588, −1.46277656236414483271641951737, 0,
1.46277656236414483271641951737, 2.34503180134762693604375030588, 2.93727634776662109083862804418, 3.86923035092990271475285611739, 4.58104496370043598564475955285, 5.46006811073047706869999706549, 6.29493150513934428120131310272, 6.68460279126830487651733874171, 7.39824793347898235416004593275
