| L(s) = 1 | − 1.80·2-s + 3-s + 1.25·4-s − 5-s − 1.80·6-s + 3.38·7-s + 1.33·8-s + 9-s + 1.80·10-s + 2.61·11-s + 1.25·12-s + 1.48·13-s − 6.11·14-s − 15-s − 4.93·16-s + 6.99·17-s − 1.80·18-s − 0.688·19-s − 1.25·20-s + 3.38·21-s − 4.72·22-s + 1.33·24-s + 25-s − 2.67·26-s + 27-s + 4.26·28-s − 5.00·29-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.577·3-s + 0.629·4-s − 0.447·5-s − 0.736·6-s + 1.28·7-s + 0.473·8-s + 0.333·9-s + 0.570·10-s + 0.789·11-s + 0.363·12-s + 0.411·13-s − 1.63·14-s − 0.258·15-s − 1.23·16-s + 1.69·17-s − 0.425·18-s − 0.157·19-s − 0.281·20-s + 0.739·21-s − 1.00·22-s + 0.273·24-s + 0.200·25-s − 0.525·26-s + 0.192·27-s + 0.806·28-s − 0.929·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.739756322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.739756322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + 2T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + 0.688T + 19T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 3.16T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 1.03T + 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 - 0.00276T + 71T^{2} \) |
| 73 | \( 1 + 0.808T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 - 0.0682T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.072772607467216557636495492608, −7.51992903799496390430958658731, −6.87878039668051249666099884001, −5.84082175046909039452770995336, −4.94432357440727346893822130300, −4.18926662092817737036935341043, −3.53333998747795516398113119118, −2.35617653216165769364689448985, −1.43590871297710868870237020220, −0.896879037654984767184035760836,
0.896879037654984767184035760836, 1.43590871297710868870237020220, 2.35617653216165769364689448985, 3.53333998747795516398113119118, 4.18926662092817737036935341043, 4.94432357440727346893822130300, 5.84082175046909039452770995336, 6.87878039668051249666099884001, 7.51992903799496390430958658731, 8.072772607467216557636495492608
