| L(s) = 1 | − 0.373·2-s + 3·3-s − 7.86·4-s − 15.6·5-s − 1.12·6-s + 5.92·8-s + 9·9-s + 5.85·10-s − 11·11-s − 23.5·12-s − 78.6·13-s − 47.0·15-s + 60.6·16-s − 53.5·17-s − 3.36·18-s − 114.·19-s + 123.·20-s + 4.10·22-s + 25.9·23-s + 17.7·24-s + 120.·25-s + 29.3·26-s + 27·27-s − 269.·29-s + 17.5·30-s − 279.·31-s − 70.0·32-s + ⋯ |
| L(s) = 1 | − 0.132·2-s + 0.577·3-s − 0.982·4-s − 1.40·5-s − 0.0762·6-s + 0.261·8-s + 0.333·9-s + 0.185·10-s − 0.301·11-s − 0.567·12-s − 1.67·13-s − 0.809·15-s + 0.947·16-s − 0.763·17-s − 0.0440·18-s − 1.38·19-s + 1.37·20-s + 0.0398·22-s + 0.235·23-s + 0.151·24-s + 0.967·25-s + 0.221·26-s + 0.192·27-s − 1.72·29-s + 0.106·30-s − 1.62·31-s − 0.386·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.08076964019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08076964019\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 0.373T + 8T^{2} \) |
| 5 | \( 1 + 15.6T + 125T^{2} \) |
| 13 | \( 1 + 78.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 269.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 60.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 131.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 617.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 478.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 134.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 580.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 317.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 282.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 658.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 374.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 103.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−9.068270254354660426124860031888, −8.189282782779648671588087571667, −7.60670640477872442377600360367, −7.05212407058980110279371496185, −5.56256997521981732084841018702, −4.51644500646808019196446929659, −4.15890415096716934850544027111, −3.15500913131323877802413985024, −1.96479996549816790572663763434, −0.12943664184585708434016183234,
0.12943664184585708434016183234, 1.96479996549816790572663763434, 3.15500913131323877802413985024, 4.15890415096716934850544027111, 4.51644500646808019196446929659, 5.56256997521981732084841018702, 7.05212407058980110279371496185, 7.60670640477872442377600360367, 8.189282782779648671588087571667, 9.068270254354660426124860031888
