L(s) = 1 | + (1.46 − 0.916i)3-s + 1.83·5-s − i·7-s + (1.31 − 2.69i)9-s + 5.57i·13-s + (2.69 − 1.68i)15-s − 6.08i·17-s + 6.93·19-s + (−0.916 − 1.46i)21-s + 0.697·23-s − 1.63·25-s + (−0.530 − 5.16i)27-s − 7.11·29-s − 1.83i·35-s + 1.87i·37-s + ⋯ |
L(s) = 1 | + (0.848 − 0.529i)3-s + 0.819·5-s − 0.377i·7-s + (0.439 − 0.898i)9-s + 1.54i·13-s + (0.695 − 0.433i)15-s − 1.47i·17-s + 1.59·19-s + (−0.200 − 0.320i)21-s + 0.145·23-s − 0.327·25-s + (−0.102 − 0.994i)27-s − 1.32·29-s − 0.309i·35-s + 0.308i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16420 - 0.794684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16420 - 0.794684i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.46 + 0.916i)T \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 1.83T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.57iT - 13T^{2} \) |
| 17 | \( 1 + 6.08iT - 17T^{2} \) |
| 19 | \( 1 - 6.93T + 19T^{2} \) |
| 23 | \( 1 - 0.697T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 - 4.69iT - 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 + 3.44T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 4.30iT - 61T^{2} \) |
| 67 | \( 1 + 4.51T + 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 11.7iT - 79T^{2} \) |
| 83 | \( 1 - 0.438iT - 83T^{2} \) |
| 89 | \( 1 - 2.64iT - 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−10.04818017648742293431346462177, −9.418545480007626557162970778803, −8.943280269034082719903967678673, −7.54869855592977136899666285318, −7.11306056421197570292106848749, −6.08215747127900844128130755984, −4.87866160431815062887566365770, −3.63560549803186206627980911089, −2.47493295539165644922585310873, −1.36287198267682676292546522789,
1.74119985657418147964017472877, 2.93727101267638402690796849013, 3.85155765413318304633554302164, 5.35518814365280949593619584118, 5.77465984234024589953729503345, 7.32437039824607383115592522612, 8.119512033134500374849484768754, 8.935950695178778810429955207399, 9.817509569507536125426689648684, 10.28848281916264552717760484104