L(s) = 1 | + (1.69 + 0.366i)3-s − 3.38i·5-s − i·7-s + (2.73 + 1.23i)9-s + 2.47·11-s − 6.19·13-s + (1.23 − 5.73i)15-s − 2.47i·17-s + 0.732i·19-s + (0.366 − 1.69i)21-s + 6.77·23-s − 6.46·25-s + (4.17 + 3.09i)27-s + 2.47i·29-s + 9.46i·31-s + ⋯ |
L(s) = 1 | + (0.977 + 0.211i)3-s − 1.51i·5-s − 0.377i·7-s + (0.910 + 0.413i)9-s + 0.747·11-s − 1.71·13-s + (0.319 − 1.48i)15-s − 0.601i·17-s + 0.167i·19-s + (0.0798 − 0.369i)21-s + 1.41·23-s − 1.29·25-s + (0.802 + 0.596i)27-s + 0.460i·29-s + 1.69i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67038 - 0.644548i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67038 - 0.644548i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.366i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 3.38iT - 5T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 6.19T + 13T^{2} \) |
| 17 | \( 1 + 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.732iT - 19T^{2} \) |
| 23 | \( 1 - 6.77T + 23T^{2} \) |
| 29 | \( 1 - 2.47iT - 29T^{2} \) |
| 31 | \( 1 - 9.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.53T + 37T^{2} \) |
| 41 | \( 1 - 9.25iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 + 8.34T + 59T^{2} \) |
| 61 | \( 1 + 4.73T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 9.25T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 - 12iT - 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 + 14.2iT - 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−11.62049526435767194644076196732, −10.18298594903472128357390401673, −9.354587936729842774715102685009, −8.848529817295963392340542561032, −7.79056441187408847968071894716, −6.91111687427186742949225834066, −4.99224634689886563230060889828, −4.55907040978619357302915374813, −3.04520749856013737828975980917, −1.36182935562868555799922961392,
2.24700403637624500328610290326, 3.05691902553690391640102960250, 4.33448054185760728079070956153, 6.08273800732047525406260384263, 7.12907362728738706438048651099, 7.60636806779634554825262437706, 8.992872723268077579726864524133, 9.727091664243783690866058905369, 10.62516491994101936575646112668, 11.68205752179577168972280404974