| L(s) = 1 | + (−1 + 1.41i)3-s − 2.82·5-s + i·7-s + (−1.00 − 2.82i)9-s − 5.65i·11-s + 4i·13-s + (2.82 − 4.00i)15-s − 2.82i·17-s + 2·19-s + (−1.41 − i)21-s + 2.82·23-s + 3.00·25-s + (5.00 + 1.41i)27-s + 5.65·29-s + (8.00 + 5.65i)33-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.816i)3-s − 1.26·5-s + 0.377i·7-s + (−0.333 − 0.942i)9-s − 1.70i·11-s + 1.10i·13-s + (0.730 − 1.03i)15-s − 0.685i·17-s + 0.458·19-s + (−0.308 − 0.218i)21-s + 0.589·23-s + 0.600·25-s + (0.962 + 0.272i)27-s + 1.05·29-s + (1.39 + 0.984i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709349 - 0.225457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709349 - 0.225457i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 2.82iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 12iT - 61T^{2} \) |
| 67 | \( 1 + 6T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.76527963464080392025405707044, −9.385609470462222636622606235133, −8.864553119411352514651599284385, −7.909559565689281395282119106355, −6.78391772049044462169822859165, −5.83165540324872493710637869314, −4.82797027588718916675367524570, −3.88280834827441278101897894534, −3.04501657203549755895224361767, −0.51899824804953233780099688543,
1.15074566330991639486068196642, 2.81487258407880701286369656536, 4.23041151525160802007333464727, 5.03221272842956234143857253410, 6.29649974716782611256341768716, 7.34831507248671814067076624446, 7.63835856636368030420013946111, 8.546332939154028911493027875620, 10.00294897059504445432387863327, 10.66528005920400787134996775452
