| L(s) = 1 | + 3-s + 5-s + 9-s − 13-s + 15-s − 2·17-s − 3·19-s − 6·23-s + 25-s + 27-s + 6·29-s + 7·31-s − 37-s − 39-s − 4·41-s − 11·43-s + 45-s + 6·47-s − 2·51-s + 4·53-s − 3·57-s + 14·59-s − 6·61-s − 65-s + 3·67-s − 6·69-s − 7·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.688·19-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.25·31-s − 0.164·37-s − 0.160·39-s − 0.624·41-s − 1.67·43-s + 0.149·45-s + 0.875·47-s − 0.280·51-s + 0.549·53-s − 0.397·57-s + 1.82·59-s − 0.768·61-s − 0.124·65-s + 0.366·67-s − 0.722·69-s − 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.82458720587933, −14.18698944687217, −13.91786185913032, −13.38656332133684, −12.88338992452073, −12.35469206087464, −11.71295258275615, −11.40056301930070, −10.36971191618044, −10.09549206538984, −9.900280809965989, −8.772460851592145, −8.718745154167864, −8.131224742505972, −7.430500520968023, −6.810694551988048, −6.345276749220524, −5.776372196763124, −4.953589785872262, −4.469875201317680, −3.859642421545060, −3.088431590272092, −2.427233754946155, −1.949724905991002, −1.084963448002560, 0,
1.084963448002560, 1.949724905991002, 2.427233754946155, 3.088431590272092, 3.859642421545060, 4.469875201317680, 4.953589785872262, 5.776372196763124, 6.345276749220524, 6.810694551988048, 7.430500520968023, 8.131224742505972, 8.718745154167864, 8.772460851592145, 9.900280809965989, 10.09549206538984, 10.36971191618044, 11.40056301930070, 11.71295258275615, 12.35469206087464, 12.88338992452073, 13.38656332133684, 13.91786185913032, 14.18698944687217, 14.82458720587933
