| L(s) = 1 | − 2i·3-s + (−1.73 − 2i)7-s − 9-s − 3.46i·11-s − 3.46·13-s + 2·19-s + (−4 + 3.46i)21-s − 3.46·23-s − 4i·27-s − 6·29-s + 8·31-s − 6.92·33-s + 2i·37-s + 6.92i·39-s − 6.92i·41-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (−0.654 − 0.755i)7-s − 0.333·9-s − 1.04i·11-s − 0.960·13-s + 0.458·19-s + (−0.872 + 0.755i)21-s − 0.722·23-s − 0.769i·27-s − 1.11·29-s + 1.43·31-s − 1.20·33-s + 0.328i·37-s + 1.10i·39-s − 1.08i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6080425297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6080425297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.73 + 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.054709315738204689292757520576, −7.53007265096617849541052536908, −6.83314508941885805193034754755, −6.24559281333466655694598349717, −5.39466073174258179683264991858, −4.26058223354259881557804255793, −3.33200182263739080228958798213, −2.40397141756496885907131939367, −1.22357759290377698896950106802, −0.19882285878926539676373456970,
1.91073604439763441405025408961, 2.90482794493001960813083249646, 3.78113112021425734016052912612, 4.69889976585551574934475989498, 5.17787798265974756094800542317, 6.15800992349820303339050792411, 7.00779257163607413579974269333, 7.80394894468831816423123065747, 8.777676153993492190905209658999, 9.504075713268961472091876733664
