| L(s) = 1 | − 2·3-s + 5-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 2·19-s + 4·23-s + 25-s + 4·27-s − 10·29-s + 4·31-s − 8·33-s + 2·37-s − 4·39-s + 12·41-s − 4·43-s + 45-s + 4·47-s − 2·53-s + 4·55-s + 4·57-s − 10·59-s + 6·61-s + 2·65-s + 4·67-s − 8·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 1.85·29-s + 0.718·31-s − 1.39·33-s + 0.328·37-s − 0.640·39-s + 1.87·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 1.30·59-s + 0.768·61-s + 0.248·65-s + 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663645753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663645753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.11648899136381, −15.53819216791032, −14.84524388208862, −14.33724069126200, −13.82759694289416, −12.98246020126257, −12.70332085274632, −11.98719987704537, −11.34370632607985, −11.09563448010884, −10.55131705046124, −9.694390529637441, −9.236758228375639, −8.688642995857843, −7.868252370906429, −7.035524807330864, −6.523352357891857, −5.975253678527251, −5.549634177328490, −4.738603035968676, −4.100231993600531, −3.338284628966599, −2.318061357481807, −1.382554821971984, −0.6510173931768425,
0.6510173931768425, 1.382554821971984, 2.318061357481807, 3.338284628966599, 4.100231993600531, 4.738603035968676, 5.549634177328490, 5.975253678527251, 6.523352357891857, 7.035524807330864, 7.868252370906429, 8.688642995857843, 9.236758228375639, 9.694390529637441, 10.55131705046124, 11.09563448010884, 11.34370632607985, 11.98719987704537, 12.70332085274632, 12.98246020126257, 13.82759694289416, 14.33724069126200, 14.84524388208862, 15.53819216791032, 16.11648899136381
