L(s) = 1 | − 2.53·2-s − 2.76·3-s + 4.40·4-s + 3.58·5-s + 6.99·6-s − 0.727·7-s − 6.09·8-s + 4.63·9-s − 9.07·10-s + 11-s − 12.1·12-s − 1.11·13-s + 1.84·14-s − 9.90·15-s + 6.60·16-s − 17-s − 11.7·18-s + 3.30·19-s + 15.7·20-s + 2.01·21-s − 2.53·22-s − 5.19·23-s + 16.8·24-s + 7.84·25-s + 2.81·26-s − 4.52·27-s − 3.20·28-s + ⋯ |
L(s) = 1 | − 1.78·2-s − 1.59·3-s + 2.20·4-s + 1.60·5-s + 2.85·6-s − 0.275·7-s − 2.15·8-s + 1.54·9-s − 2.86·10-s + 0.301·11-s − 3.51·12-s − 0.308·13-s + 0.492·14-s − 2.55·15-s + 1.65·16-s − 0.242·17-s − 2.76·18-s + 0.758·19-s + 3.53·20-s + 0.438·21-s − 0.539·22-s − 1.08·23-s + 3.43·24-s + 1.56·25-s + 0.551·26-s − 0.870·27-s − 0.606·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4754759868\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4754759868\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 2.53T + 2T^{2} \) |
| 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 + 0.727T + 7T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 19 | \( 1 - 3.30T + 19T^{2} \) |
| 23 | \( 1 + 5.19T + 23T^{2} \) |
| 29 | \( 1 + 1.68T + 29T^{2} \) |
| 31 | \( 1 - 3.36T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 47 | \( 1 - 2.08T + 47T^{2} \) |
| 53 | \( 1 + 0.317T + 53T^{2} \) |
| 59 | \( 1 - 5.62T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 7.90T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 8.85T + 83T^{2} \) |
| 89 | \( 1 - 7.04T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.81736706262290370990305735585, −6.93732713929872800850616474472, −6.59850296281730378821993105406, −5.95756806315223074772880978725, −5.47999863091654182965485620489, −4.62888026706525845763516485745, −3.13791441962179468292770801024, −2.00238531699700326353166854651, −1.51436026579374727142849131038, −0.50507324122225754613779752566,
0.50507324122225754613779752566, 1.51436026579374727142849131038, 2.00238531699700326353166854651, 3.13791441962179468292770801024, 4.62888026706525845763516485745, 5.47999863091654182965485620489, 5.95756806315223074772880978725, 6.59850296281730378821993105406, 6.93732713929872800850616474472, 7.81736706262290370990305735585