| L(s) = 1 | + (1.68 + 0.420i)3-s + 3.91i·5-s + 2.64·7-s + (2.64 + 1.41i)9-s − 4.55·13-s + (−1.64 + 6.57i)15-s + 0.979·19-s + (4.44 + 1.11i)21-s − 7.48i·23-s − 10.2·25-s + (3.85 + 3.48i)27-s + 10.3i·35-s + (−7.64 − 1.91i)39-s + (−5.53 + 10.3i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | + (0.970 + 0.242i)3-s + 1.74i·5-s + 0.999·7-s + (0.881 + 0.471i)9-s − 1.26·13-s + (−0.424 + 1.69i)15-s + 0.224·19-s + (0.970 + 0.242i)21-s − 1.56i·23-s − 2.05·25-s + (0.740 + 0.671i)27-s + 1.74i·35-s + (−1.22 − 0.306i)39-s + (−0.824 + 1.54i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73764 + 1.35606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73764 + 1.35606i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.68 - 0.420i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 3.91iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4.55T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 0.979T + 19T^{2} \) |
| 23 | \( 1 + 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.29iT - 59T^{2} \) |
| 61 | \( 1 - 15.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.45840203041725445400998995148, −10.06918589303301743692671609726, −8.940863910668026927814286591398, −7.914642704423832105932547280529, −7.34036259128628844401006887119, −6.53185768551640714996107135553, −5.04855765382418921588430452577, −4.01316831370351382276178475354, −2.81776403415260052032449934761, −2.17100690012923778954199436275,
1.17334719512709836318321605632, 2.20222838837364690325138536062, 3.84979919738859019654083551106, 4.81242895810211097907118851193, 5.45412528010638925516366488210, 7.17343353390264360373710797556, 7.928107798311165645940676587496, 8.509006867165859484039113312438, 9.374319352203642518330110774395, 9.888331505872701041682295387125
