| L(s) = 1 | + (1 + 2i)5-s − i·7-s − 6·11-s + 2i·13-s − 2i·17-s + 4·19-s − 4i·23-s + (−3 + 4i)25-s − 2·29-s + 2·31-s + (2 − i)35-s + 10i·37-s + 6·41-s + 2i·43-s − 2i·47-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.894i)5-s − 0.377i·7-s − 1.80·11-s + 0.554i·13-s − 0.485i·17-s + 0.917·19-s − 0.834i·23-s + (−0.600 + 0.800i)25-s − 0.371·29-s + 0.359·31-s + (0.338 − 0.169i)35-s + 1.64i·37-s + 0.937·41-s + 0.304i·43-s − 0.291i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 12T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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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
−7.70852976726498004478214867713, −7.32211285711863986699138245120, −6.53013539327184428268540906683, −5.77714711709376559485511397894, −5.03929792565959955871828392673, −4.27575697998073753705432340931, −2.96871845110620378089239859152, −2.76573844523594686552095099643, −1.54801182283992455520555027988, 0,
1.28484344881747618300901556250, 2.34786738567821452889483514062, 3.07811329672989732029263648292, 4.20541565733904791844462749356, 5.09669387133980909337332853989, 5.59830083859186390637075584103, 6.02346289337222199897684901664, 7.47663100097631047994175933106, 7.72644919496740081836875715990
