| L(s) = 1 | + (−1.68 + 0.420i)3-s + 3.91i·5-s + 2.64·7-s + (2.64 − 1.41i)9-s + 4.55·13-s + (−1.64 − 6.57i)15-s − 0.979·19-s + (−4.44 + 1.11i)21-s + 7.48i·23-s − 10.2·25-s + (−3.85 + 3.48i)27-s + 10.3i·35-s + (−7.64 + 1.91i)39-s + (5.53 + 10.3i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | + (−0.970 + 0.242i)3-s + 1.74i·5-s + 0.999·7-s + (0.881 − 0.471i)9-s + 1.26·13-s + (−0.424 − 1.69i)15-s − 0.224·19-s + (−0.970 + 0.242i)21-s + 1.56i·23-s − 2.05·25-s + (−0.740 + 0.671i)27-s + 1.74i·35-s + (−1.22 + 0.306i)39-s + (0.824 + 1.54i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.738547 + 0.946365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738547 + 0.946365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 - 0.420i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 3.91iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.55T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 0.979T + 19T^{2} \) |
| 23 | \( 1 - 7.48iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 15.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + 18.1iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.82431474004373346906507507220, −10.31064450135461846968293814331, −9.204917708504031229217768279270, −7.87863029476685535429762618045, −7.13473597749149487891961406127, −6.23674309709388288722127386404, −5.54983087557144758789316878362, −4.22639913326293461649867053375, −3.26715157104950490746144847682, −1.64583739517199330391361523104,
0.805570303058247100850503369612, 1.77353474042231410817815366885, 4.16690813049252727789207260154, 4.78881329626783200155910522693, 5.59604661597124071762051101105, 6.48255083847402487712705957805, 7.86243865026855241795626585973, 8.447291696539952354851100447026, 9.239504675215700898218777183937, 10.49595695147299199078584153832
