L(s) = 1 | − 1.36·2-s − 3-s − 0.145·4-s + 2.48·5-s + 1.36·6-s + 7-s + 2.92·8-s + 9-s − 3.38·10-s − 2.24·11-s + 0.145·12-s − 4.63·13-s − 1.36·14-s − 2.48·15-s − 3.68·16-s − 2.62·17-s − 1.36·18-s − 4.04·19-s − 0.361·20-s − 21-s + 3.05·22-s − 0.188·23-s − 2.92·24-s + 1.17·25-s + 6.31·26-s − 27-s − 0.145·28-s + ⋯ |
L(s) = 1 | − 0.962·2-s − 0.577·3-s − 0.0727·4-s + 1.11·5-s + 0.555·6-s + 0.377·7-s + 1.03·8-s + 0.333·9-s − 1.07·10-s − 0.677·11-s + 0.0419·12-s − 1.28·13-s − 0.363·14-s − 0.641·15-s − 0.921·16-s − 0.635·17-s − 0.320·18-s − 0.927·19-s − 0.0808·20-s − 0.218·21-s + 0.651·22-s − 0.0392·23-s − 0.596·24-s + 0.235·25-s + 1.23·26-s − 0.192·27-s − 0.0274·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 5 | \( 1 - 2.48T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + 4.04T + 19T^{2} \) |
| 23 | \( 1 + 0.188T + 23T^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 + 1.95T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 8.87T + 53T^{2} \) |
| 59 | \( 1 - 5.06T + 59T^{2} \) |
| 61 | \( 1 - 4.96T + 61T^{2} \) |
| 67 | \( 1 - 6.69T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 8.96T + 89T^{2} \) |
| 97 | \( 1 - 2.38T + 97T^{2} \) |
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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
−7.71308511597534436565597313262, −6.73664655964697943045187099359, −6.29631222410158909698998217394, −5.28388123835501259960564841967, −4.82088741064874155055777326479, −4.22871143228438848790618643482, −2.60076600485142014496308750906, −2.11201148674487302109123225247, −1.06378363907496017162544935661, 0,
1.06378363907496017162544935661, 2.11201148674487302109123225247, 2.60076600485142014496308750906, 4.22871143228438848790618643482, 4.82088741064874155055777326479, 5.28388123835501259960564841967, 6.29631222410158909698998217394, 6.73664655964697943045187099359, 7.71308511597534436565597313262