| L(s) = 1 | − 1.41i·3-s + (1.73 + 1.41i)5-s − 2.44i·7-s + 0.999·9-s + 3.46·11-s + (2.00 − 2.44i)15-s + 4.89i·17-s + 3.46·19-s − 3.46·21-s + 2.44i·23-s + (0.999 + 4.89i)25-s − 5.65i·27-s − 4·31-s − 4.89i·33-s + (3.46 − 4.24i)35-s + ⋯ |
| L(s) = 1 | − 0.816i·3-s + (0.774 + 0.632i)5-s − 0.925i·7-s + 0.333·9-s + 1.04·11-s + (0.516 − 0.632i)15-s + 1.18i·17-s + 0.794·19-s − 0.755·21-s + 0.510i·23-s + (0.199 + 0.979i)25-s − 1.08i·27-s − 0.718·31-s − 0.852i·33-s + (0.585 − 0.717i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.177405543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.177405543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| good | 3 | \( 1 + 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 7.34iT - 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−9.642011668823009907374866635380, −8.846542182029182471528032692662, −7.52680694396134923847010432784, −7.24148655510805656267274271378, −6.35981304926624898590321313384, −5.70909742931164824202641692653, −4.25217975103964375793307081218, −3.43975238087377670073614455875, −1.97866523500807962753182022751, −1.18374643757067192353577345042,
1.27876517353509605224250824440, 2.55549500052085258332401945692, 3.75664049249258193904455512269, 4.83066084607386694877160437208, 5.32395816591028880136917080310, 6.32603436921949781356015901397, 7.20831380421221254487300371176, 8.537667652895850764153835331988, 9.094005497951421004726410253082, 9.688072151900944528659994573432
