| L(s) = 1 | − 2-s + 2.37·3-s + 4-s + 3.73·5-s − 2.37·6-s + 1.22·7-s − 8-s + 2.63·9-s − 3.73·10-s − 1.36·11-s + 2.37·12-s − 0.639·13-s − 1.22·14-s + 8.86·15-s + 16-s + 0.0658·17-s − 2.63·18-s − 0.565·19-s + 3.73·20-s + 2.91·21-s + 1.36·22-s + 23-s − 2.37·24-s + 8.94·25-s + 0.639·26-s − 0.857·27-s + 1.22·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.37·3-s + 0.5·4-s + 1.67·5-s − 0.969·6-s + 0.463·7-s − 0.353·8-s + 0.879·9-s − 1.18·10-s − 0.410·11-s + 0.685·12-s − 0.177·13-s − 0.328·14-s + 2.28·15-s + 0.250·16-s + 0.0159·17-s − 0.621·18-s − 0.129·19-s + 0.835·20-s + 0.636·21-s + 0.290·22-s + 0.208·23-s − 0.484·24-s + 1.78·25-s + 0.125·26-s − 0.165·27-s + 0.231·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.607380648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.607380648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.37T + 3T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 + 0.639T + 13T^{2} \) |
| 17 | \( 1 - 0.0658T + 17T^{2} \) |
| 19 | \( 1 + 0.565T + 19T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 + 9.44T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 - 1.77T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 - 7.16T + 47T^{2} \) |
| 53 | \( 1 - 8.45T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 2.82T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 4.66T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.278583999551625190382153680629, −7.55364444837399062789334415518, −6.87182027158207062567848708544, −5.98392391660311784521058214427, −5.37775444384323636898540856972, −4.39840749349841779829204845461, −3.23137281416017541773676929251, −2.43909602909523389506568949559, −2.07984979666867604589579895846, −1.09458717286490883938766221733,
1.09458717286490883938766221733, 2.07984979666867604589579895846, 2.43909602909523389506568949559, 3.23137281416017541773676929251, 4.39840749349841779829204845461, 5.37775444384323636898540856972, 5.98392391660311784521058214427, 6.87182027158207062567848708544, 7.55364444837399062789334415518, 8.278583999551625190382153680629
