| L(s) = 1 | + (−0.707 − 1.58i)3-s − 1.41i·5-s − i·7-s + (−2.00 + 2.23i)9-s − 4.47·11-s − 0.837·13-s + (−2.23 + 1.00i)15-s + 1.64i·17-s − 7.16i·19-s + (−1.58 + 0.707i)21-s − 5.65·23-s + 2.99·25-s + (4.94 + 1.58i)27-s + 7.30i·29-s + 6.32i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 − 0.912i)3-s − 0.632i·5-s − 0.377i·7-s + (−0.666 + 0.745i)9-s − 1.34·11-s − 0.232·13-s + (−0.577 + 0.258i)15-s + 0.398i·17-s − 1.64i·19-s + (−0.345 + 0.154i)21-s − 1.17·23-s + 0.599·25-s + (0.952 + 0.304i)27-s + 1.35i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0800202 + 0.433753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0800202 + 0.433753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 1.58i)T \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 0.837T + 13T^{2} \) |
| 17 | \( 1 - 1.64iT - 17T^{2} \) |
| 19 | \( 1 + 7.16iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 7.30iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 + 1.18iT - 41T^{2} \) |
| 43 | \( 1 + 4.32iT - 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 3.16T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 + 8.32T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 10.1iT - 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−10.30535829582622035865303761841, −8.896620463884719959563806982208, −8.257958441526784122047150481944, −7.30449584685710036168309395179, −6.62358979215371311400367889524, −5.30129247837565988254854679764, −4.85901847032516113702176505028, −3.09634837561427023532095664143, −1.78823754687856873557089327133, −0.23209573272376829836668012719,
2.40592926154921393596040310944, 3.47155215969148118065310607293, 4.60358317178487391461030322792, 5.62932824361121677814772091516, 6.24913645628828298403415881350, 7.63246036889221454012008793991, 8.340313881283750945940905873016, 9.611781333785276167506466341095, 10.15020137951685541006769816361, 10.78986486339738787059160821050
