| L(s) = 1 | + 2·3-s + (−2 − i)5-s + (3 − 3i)7-s + 9-s + (1 + i)11-s − 2i·13-s + (−4 − 2i)15-s + (1 − i)17-s + (3 + 3i)19-s + (6 − 6i)21-s + (1 + i)23-s + (3 + 4i)25-s − 4·27-s + (−7 + 7i)29-s − 2i·31-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + (−0.894 − 0.447i)5-s + (1.13 − 1.13i)7-s + 0.333·9-s + (0.301 + 0.301i)11-s − 0.554i·13-s + (−1.03 − 0.516i)15-s + (0.242 − 0.242i)17-s + (0.688 + 0.688i)19-s + (1.30 − 1.30i)21-s + (0.208 + 0.208i)23-s + (0.600 + 0.800i)25-s − 0.769·27-s + (−1.29 + 1.29i)29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68941 - 0.545382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68941 - 0.545382i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 + (-3 + 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1 - i)T + 11iT^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-1 + i)T - 17iT^{2} \) |
| 19 | \( 1 + (-3 - 3i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1 - i)T + 23iT^{2} \) |
| 29 | \( 1 + (7 - 7i)T - 29iT^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + (-7 - 7i)T + 47iT^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 + (3 - 3i)T - 59iT^{2} \) |
| 61 | \( 1 + (1 + i)T + 61iT^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (11 - 11i)T - 97iT^{2} \) |
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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
−11.49078568171744158644264814125, −10.68888766525442977472543815130, −9.478182554347819723251539097819, −8.545716628297638734549363976848, −7.69993671153038351534142081142, −7.35378707532596062831670289749, −5.29173233385468064927013894560, −4.15979097804442808367545409375, −3.30341074745633033539460164973, −1.40955617353133347827143458429,
2.11640262027913904639651854572, 3.22925393380734205009349706187, 4.42050212556708163117585892987, 5.77524046318002532213843835306, 7.27628014750412082123667030193, 8.085843928995543131546044172507, 8.742467991714931292882188491978, 9.528214016195289226326838627861, 11.18527061447747119851428558245, 11.50814938353155228541648729155
