| L(s) = 1 | + (1.61 − 0.618i)3-s + 3.23·5-s + i·7-s + (2.23 − 2.00i)9-s − 4.47i·11-s + 1.23i·13-s + (5.23 − 2.00i)15-s + 2i·17-s − 7.23·19-s + (0.618 + 1.61i)21-s + 0.472·23-s + 5.47·25-s + (2.38 − 4.61i)27-s + 8.94i·31-s + (−2.76 − 7.23i)33-s + ⋯ |
| L(s) = 1 | + (0.934 − 0.356i)3-s + 1.44·5-s + 0.377i·7-s + (0.745 − 0.666i)9-s − 1.34i·11-s + 0.342i·13-s + (1.35 − 0.516i)15-s + 0.485i·17-s − 1.66·19-s + (0.134 + 0.353i)21-s + 0.0984·23-s + 1.09·25-s + (0.458 − 0.888i)27-s + 1.60i·31-s + (−0.481 − 1.25i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48445 - 0.530236i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48445 - 0.530236i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.61 + 0.618i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 6.47iT - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 0.472T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 1.23iT - 59T^{2} \) |
| 61 | \( 1 + 11.7iT - 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 6.94T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 4.94iT - 79T^{2} \) |
| 83 | \( 1 - 13.2iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 3.52T + 97T^{2} \) |
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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
−10.31029224067018883100271771262, −9.425651115152558422677236564171, −8.695002088354570282249466023578, −8.188879400934744408445109076609, −6.59919948266156650455513040904, −6.26274766130226540521493806937, −5.06075724820544981892478111924, −3.59011774973407130960702526833, −2.50391266319165220192892077472, −1.54972273034918846143532238428,
1.85058141211833865736529027809, 2.55494884535674399791079643881, 4.08642522780298187312639876311, 4.92841506400247121958072650782, 6.12903207696050279927386826633, 7.10051262179613694444387755747, 8.014153572995319574825864188822, 9.088340961266856712683849212972, 9.730130870992550678988436364640, 10.20874509019846272107356349447
