| 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
−10.28848281916264552717760484104, −9.817509569507536125426689648684, −8.935950695178778810429955207399, −8.119512033134500374849484768754, −7.32437039824607383115592522612, −5.77465984234024589953729503345, −5.35518814365280949593619584118, −3.85155765413318304633554302164, −2.93727101267638402690796849013, −1.74119985657418147964017472877,
1.36287198267682676292546522789, 2.47493295539165644922585310873, 3.63560549803186206627980911089, 4.87866160431815062887566365770, 6.08215747127900844128130755984, 7.11306056421197570292106848749, 7.54869855592977136899666285318, 8.943280269034082719903967678673, 9.418545480007626557162970778803, 10.04818017648742293431346462177
