| L(s) = 1 | + (−1 + 1.41i)3-s − 1.41i·5-s + (2 − 1.73i)7-s + (−1.00 − 2.82i)9-s − 2.44·11-s − 2·13-s + (2.00 + 1.41i)15-s − 7.34·17-s − 4·19-s + (0.449 + 4.56i)21-s − 1.41i·23-s + 2.99·25-s + (5.00 + 1.41i)27-s − 4.89·29-s − 6.92i·31-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.816i)3-s − 0.632i·5-s + (0.755 − 0.654i)7-s + (−0.333 − 0.942i)9-s − 0.738·11-s − 0.554·13-s + (0.516 + 0.365i)15-s − 1.78·17-s − 0.917·19-s + (0.0980 + 0.995i)21-s − 0.294i·23-s + 0.599·25-s + (0.962 + 0.272i)27-s − 0.909·29-s − 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.313410 - 0.486047i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.313410 - 0.486047i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 + 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.44T + 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.65iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−10.45488090707753373306003474172, −9.353485876253744424766740499141, −8.678376466089997931952504773412, −7.65802342051279478465501890688, −6.58567532582321219593683173703, −5.45083263184198956389684435492, −4.61418753647060352570556274699, −4.07162899147964540471787764884, −2.26583491414151591611899055105, −0.30906461806526161905850965814,
1.89631661981466758441758389026, 2.73786567729869849639963742094, 4.62319015998263398936610496487, 5.37416322002163886508328774560, 6.50301155224613265292611019935, 7.09133137766045022230643418515, 8.151008684240010580980692345924, 8.788911147777769775350153484870, 10.18364117615456387355737975760, 11.05285401382040822109614301131
