| L(s) = 1 | − 2·3-s − 5-s + 9-s + 3·11-s + 13-s + 2·15-s − 6·17-s − 19-s − 9·23-s + 25-s + 4·27-s − 6·29-s − 8·31-s − 6·33-s + 7·37-s − 2·39-s + 3·41-s + 2·43-s − 45-s − 9·47-s + 12·51-s − 9·53-s − 3·55-s + 2·57-s − 8·61-s − 65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s − 0.229·19-s − 1.87·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 1.04·33-s + 1.15·37-s − 0.320·39-s + 0.468·41-s + 0.304·43-s − 0.149·45-s − 1.31·47-s + 1.68·51-s − 1.23·53-s − 0.404·55-s + 0.264·57-s − 1.02·61-s − 0.124·65-s + 0.977·67-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
| \(L(1)\) |
\(\approx\) |
\(0.3997057647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3997057647\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.16917055030531, −15.61229512057653, −14.84817153486539, −14.45994585898405, −13.75370014370163, −13.05107373778488, −12.54168784979543, −11.98063062207968, −11.39622501590575, −11.06515206966507, −10.67523446714432, −9.647527444903487, −9.292681749591571, −8.505245831673863, −7.898362053791192, −7.170595979024867, −6.402866844209883, −6.165516163958895, −5.455147481837667, −4.597210231615900, −4.119357871052981, −3.465587643336563, −2.272621949206861, −1.509013855139809, −0.2931181839683186,
0.2931181839683186, 1.509013855139809, 2.272621949206861, 3.465587643336563, 4.119357871052981, 4.597210231615900, 5.455147481837667, 6.165516163958895, 6.402866844209883, 7.170595979024867, 7.898362053791192, 8.505245831673863, 9.292681749591571, 9.647527444903487, 10.67523446714432, 11.06515206966507, 11.39622501590575, 11.98063062207968, 12.54168784979543, 13.05107373778488, 13.75370014370163, 14.45994585898405, 14.84817153486539, 15.61229512057653, 16.16917055030531
