| L(s) = 1 | + i·3-s + 3.41i·5-s − 1.41·7-s − 9-s + 4.82i·11-s + 0.828i·13-s − 3.41·15-s + 4.82·17-s + 2.82i·19-s − 1.41i·21-s + 1.17·23-s − 6.65·25-s − i·27-s − 7.41i·29-s − 7.07·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.52i·5-s − 0.534·7-s − 0.333·9-s + 1.45i·11-s + 0.229i·13-s − 0.881·15-s + 1.17·17-s + 0.648i·19-s − 0.308i·21-s + 0.244·23-s − 1.33·25-s − 0.192i·27-s − 1.37i·29-s − 1.27·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195083658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195083658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.41iT - 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 4.82iT - 11T^{2} \) |
| 13 | \( 1 - 0.828iT - 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 + 7.41iT - 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 6.82iT - 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 1.75iT - 53T^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 + 0.343iT - 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 + 8.82iT - 83T^{2} \) |
| 89 | \( 1 + 5.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−9.757392053525104553993276012944, −9.582846050266060011169003379124, −8.075639232466104779992337273073, −7.39828127940770047453451101130, −6.63008454217086232538018307980, −5.91652620046539319883213307928, −4.79199220276005773722625936669, −3.74527519134794824439130655578, −3.05894501345611092851825850838, −1.98688316324934969795210808010,
0.48691453854691356156829063829, 1.38356084350005704101408452395, 2.95040778988887081601800938075, 3.84413931797163736311661295286, 5.20013157075834892367087682760, 5.59351933969330532979706683902, 6.55419877521952211605275077544, 7.65510154907028809304158762247, 8.259182407742600551818469106126, 9.104677397577481342880717631773
