L(s) = 1 | − 2-s − 1.47·3-s + 4-s + 2.55·5-s + 1.47·6-s − 3.54·7-s − 8-s − 0.817·9-s − 2.55·10-s − 0.373·11-s − 1.47·12-s − 2.41·13-s + 3.54·14-s − 3.77·15-s + 16-s − 0.178·17-s + 0.817·18-s + 6.41·19-s + 2.55·20-s + 5.23·21-s + 0.373·22-s + 23-s + 1.47·24-s + 1.51·25-s + 2.41·26-s + 5.63·27-s − 3.54·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.852·3-s + 0.5·4-s + 1.14·5-s + 0.603·6-s − 1.33·7-s − 0.353·8-s − 0.272·9-s − 0.807·10-s − 0.112·11-s − 0.426·12-s − 0.668·13-s + 0.946·14-s − 0.973·15-s + 0.250·16-s − 0.0431·17-s + 0.192·18-s + 1.47·19-s + 0.570·20-s + 1.14·21-s + 0.0796·22-s + 0.208·23-s + 0.301·24-s + 0.303·25-s + 0.472·26-s + 1.08·27-s − 0.669·28-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
\(L(1)\) |
\(\approx\) |
\(0.7398359458\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7398359458\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.47T + 3T^{2} \) |
| 5 | \( 1 - 2.55T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 0.373T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 + 0.178T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 - 4.56T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 41 | \( 1 + 9.99T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 3.45T + 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 + 1.13T + 61T^{2} \) |
| 67 | \( 1 + 3.19T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 1.68T + 79T^{2} \) |
| 83 | \( 1 - 4.76T + 83T^{2} \) |
| 89 | \( 1 + 7.78T + 89T^{2} \) |
| 97 | \( 1 - 1.11T + 97T^{2} \) |
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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
−8.170550756515625890049637217020, −7.15884073044916516583872951463, −6.48708523224400440104104223308, −6.21417764921899076266041673059, −5.34254089731495820488428302432, −4.83126514282401873979876627992, −3.17577442412494463591181071610, −2.85968672763566731821537172661, −1.63223523923147217851837805423, −0.52262054811226271474293860907,
0.52262054811226271474293860907, 1.63223523923147217851837805423, 2.85968672763566731821537172661, 3.17577442412494463591181071610, 4.83126514282401873979876627992, 5.34254089731495820488428302432, 6.21417764921899076266041673059, 6.48708523224400440104104223308, 7.15884073044916516583872951463, 8.170550756515625890049637217020