| L(s) = 1 | − 2-s − 0.987·3-s + 4-s − 4.26·5-s + 0.987·6-s + 0.584·7-s − 8-s − 2.02·9-s + 4.26·10-s + 6.24·11-s − 0.987·12-s − 6.01·13-s − 0.584·14-s + 4.20·15-s + 16-s − 4.53·17-s + 2.02·18-s − 6.84·19-s − 4.26·20-s − 0.577·21-s − 6.24·22-s + 23-s + 0.987·24-s + 13.1·25-s + 6.01·26-s + 4.96·27-s + 0.584·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.570·3-s + 0.5·4-s − 1.90·5-s + 0.403·6-s + 0.221·7-s − 0.353·8-s − 0.674·9-s + 1.34·10-s + 1.88·11-s − 0.285·12-s − 1.66·13-s − 0.156·14-s + 1.08·15-s + 0.250·16-s − 1.10·17-s + 0.477·18-s − 1.56·19-s − 0.952·20-s − 0.126·21-s − 1.33·22-s + 0.208·23-s + 0.201·24-s + 2.63·25-s + 1.18·26-s + 0.954·27-s + 0.110·28-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
| \(L(1)\) |
\(\approx\) |
\(0.07118296837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07118296837\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.987T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 7 | \( 1 - 0.584T + 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 + 6.01T + 13T^{2} \) |
| 17 | \( 1 + 4.53T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 + 9.15T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 8.62T + 43T^{2} \) |
| 47 | \( 1 + 6.91T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 - 2.42T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.39T + 73T^{2} \) |
| 79 | \( 1 + 6.20T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 - 5.91T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.167614182069516244411487995953, −7.34537593596405427590909256818, −6.72691649127767566051974499394, −6.39424774763939977852539521634, −4.95388735410429047137748731946, −4.49976925473115564566212562614, −3.70068559662144512096997505588, −2.79299869609347903542514335627, −1.59097069709069978480868511213, −0.15925356558650919615288408502,
0.15925356558650919615288408502, 1.59097069709069978480868511213, 2.79299869609347903542514335627, 3.70068559662144512096997505588, 4.49976925473115564566212562614, 4.95388735410429047137748731946, 6.39424774763939977852539521634, 6.72691649127767566051974499394, 7.34537593596405427590909256818, 8.167614182069516244411487995953
