| L(s) = 1 | + 2·3-s − 5-s + 9-s − 3·11-s + 5·13-s − 2·15-s + 6·17-s + 19-s − 3·23-s + 25-s − 4·27-s − 6·29-s + 4·31-s − 6·33-s + 11·37-s + 10·39-s + 3·41-s + 10·43-s − 45-s − 3·47-s + 12·51-s + 3·53-s + 3·55-s + 2·57-s − 4·61-s − 5·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.38·13-s − 0.516·15-s + 1.45·17-s + 0.229·19-s − 0.625·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 0.718·31-s − 1.04·33-s + 1.80·37-s + 1.60·39-s + 0.468·41-s + 1.52·43-s − 0.149·45-s − 0.437·47-s + 1.68·51-s + 0.412·53-s + 0.404·55-s + 0.264·57-s − 0.512·61-s − 0.620·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.727587534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.727587534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.312353373911425463378423277597, −7.81189036994320081381812287447, −7.47025394478870762896015822967, −6.09431538082891075849115819441, −5.65372048903059002115277189268, −4.45082680945347931104217752177, −3.63096321073868899269924714815, −3.08375678221313543410500442005, −2.17787868933866687584737323067, −0.919024052519317686191599815600,
0.919024052519317686191599815600, 2.17787868933866687584737323067, 3.08375678221313543410500442005, 3.63096321073868899269924714815, 4.45082680945347931104217752177, 5.65372048903059002115277189268, 6.09431538082891075849115819441, 7.47025394478870762896015822967, 7.81189036994320081381812287447, 8.312353373911425463378423277597
