| L(s) = 1 | − 2.53·3-s + 3.24i·5-s − i·7-s + 3.43·9-s − 5.56i·11-s + (1.56 + 3.24i)13-s − 8.24i·15-s − 3.12·17-s + 6.56i·19-s + 2.53i·21-s − 0.712·23-s − 5.56·25-s − 1.11·27-s + 5.43·29-s + i·31-s + ⋯ |
| L(s) = 1 | − 1.46·3-s + 1.45i·5-s − 0.377i·7-s + 1.14·9-s − 1.67i·11-s + (0.433 + 0.901i)13-s − 2.12i·15-s − 0.757·17-s + 1.50i·19-s + 0.553i·21-s − 0.148·23-s − 1.11·25-s − 0.214·27-s + 1.00·29-s + 0.179i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7419295080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7419295080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-1.56 - 3.24i)T \) |
| good | 3 | \( 1 + 2.53T + 3T^{2} \) |
| 5 | \( 1 - 3.24iT - 5T^{2} \) |
| 11 | \( 1 + 5.56iT - 11T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 6.56iT - 19T^{2} \) |
| 23 | \( 1 + 0.712T + 23T^{2} \) |
| 29 | \( 1 - 5.43T + 29T^{2} \) |
| 31 | \( 1 - iT - 31T^{2} \) |
| 37 | \( 1 + 3.96iT - 37T^{2} \) |
| 41 | \( 1 + 2.53iT - 41T^{2} \) |
| 43 | \( 1 - 8.32T + 43T^{2} \) |
| 47 | \( 1 + 12.3iT - 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 - 15.1iT - 59T^{2} \) |
| 61 | \( 1 + 0.438T + 61T^{2} \) |
| 67 | \( 1 + 0.684iT - 67T^{2} \) |
| 71 | \( 1 - 6.24iT - 71T^{2} \) |
| 73 | \( 1 - 4.36iT - 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 + 3.43iT - 83T^{2} \) |
| 89 | \( 1 - 3.24iT - 89T^{2} \) |
| 97 | \( 1 - 10.8iT - 97T^{2} \) |
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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
−8.981701126518274123897902648006, −8.187520823152477952227552852874, −7.10261902974487027215458529382, −6.64400989601956514209621369731, −5.97833041235345898784986829966, −5.54127057420299277096559638895, −4.17927956842751480009420192510, −3.59807677147832723135155233324, −2.43396055395016958345851530630, −0.982645829238516091867722364468,
0.37914971874267199046495006266, 1.34267567248115891921485806462, 2.59239040196728545824296441131, 4.33612765074321233732764568446, 4.76570038207136753326563098375, 5.24826912273802646805458110083, 6.14425114437537917198617096961, 6.80948961351815018051706912201, 7.74049086647177661093642849395, 8.598900481416229105783716701424
