L(s) = 1 | + (1.24 − 0.678i)2-s + (1.07 − 1.68i)4-s + (−0.920 + 0.920i)5-s − 4.02i·7-s + (0.196 − 2.82i)8-s + (−0.517 + 1.76i)10-s + (0.00588 − 0.00588i)11-s + (−1.05 − 1.05i)13-s + (−2.72 − 4.98i)14-s + (−1.67 − 3.63i)16-s + 8.16·17-s + (−2.21 − 2.21i)19-s + (0.556 + 2.54i)20-s + (0.00330 − 0.0112i)22-s + 4.29i·23-s + ⋯ |
L(s) = 1 | + (0.877 − 0.479i)2-s + (0.539 − 0.841i)4-s + (−0.411 + 0.411i)5-s − 1.51i·7-s + (0.0695 − 0.997i)8-s + (−0.163 + 0.558i)10-s + (0.00177 − 0.00177i)11-s + (−0.291 − 0.291i)13-s + (−0.729 − 1.33i)14-s + (−0.417 − 0.908i)16-s + 1.97·17-s + (−0.509 − 0.509i)19-s + (0.124 + 0.568i)20-s + (0.000705 − 0.00240i)22-s + 0.895i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0381 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0381 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52794 - 1.47075i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52794 - 1.47075i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.24 + 0.678i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.920 - 0.920i)T - 5iT^{2} \) |
| 7 | \( 1 + 4.02iT - 7T^{2} \) |
| 11 | \( 1 + (-0.00588 + 0.00588i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.05 + 1.05i)T + 13iT^{2} \) |
| 17 | \( 1 - 8.16T + 17T^{2} \) |
| 19 | \( 1 + (2.21 + 2.21i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.29iT - 23T^{2} \) |
| 29 | \( 1 + (1.74 + 1.74i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 + (3.07 - 3.07i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.43iT - 41T^{2} \) |
| 43 | \( 1 + (-6.55 + 6.55i)T - 43iT^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + (8.65 - 8.65i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.98 - 6.98i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.243 + 0.243i)T + 61iT^{2} \) |
| 67 | \( 1 + (-4.55 - 4.55i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 7.35iT - 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 + (-8.93 - 8.93i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.17iT - 89T^{2} \) |
| 97 | \( 1 - 3.27T + 97T^{2} \) |
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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
−10.96024963081187417084621532826, −10.31082109534328098703102748338, −9.572376145299620580019378865434, −7.72715979302660675126743306063, −7.27130901415841174805327307145, −6.09422877214553543262785678930, −4.91577858123587753295682767949, −3.85144298260746023688143455644, −3.07308885334030755256282683766, −1.13943247415978004736359037036,
2.26833808727344750928150620302, 3.49183532012560425921108585561, 4.76403148519037754229182681039, 5.60835549440267499935759843721, 6.42260885059104412083933514330, 7.77796746424565019831626505216, 8.379151207609078613063931451498, 9.356888388123584234579921243195, 10.66509973689270380007701695366, 11.94720421800633914226559164156