L(s) = 1 | − 0.811·2-s + 3-s − 1.34·4-s − 5-s − 0.811·6-s + 2.71·8-s + 9-s + 0.811·10-s + 11-s − 1.34·12-s − 5.06·13-s − 15-s + 0.479·16-s − 0.968·17-s − 0.811·18-s − 2.68·19-s + 1.34·20-s − 0.811·22-s − 5.21·23-s + 2.71·24-s + 25-s + 4.11·26-s + 27-s − 4.74·29-s + 0.811·30-s − 0.0517·31-s − 5.81·32-s + ⋯ |
L(s) = 1 | − 0.574·2-s + 0.577·3-s − 0.670·4-s − 0.447·5-s − 0.331·6-s + 0.958·8-s + 0.333·9-s + 0.256·10-s + 0.301·11-s − 0.387·12-s − 1.40·13-s − 0.258·15-s + 0.119·16-s − 0.234·17-s − 0.191·18-s − 0.615·19-s + 0.299·20-s − 0.173·22-s − 1.08·23-s + 0.553·24-s + 0.200·25-s + 0.807·26-s + 0.192·27-s − 0.881·29-s + 0.148·30-s − 0.00928·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8155970124\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8155970124\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 0.811T + 2T^{2} \) |
| 13 | \( 1 + 5.06T + 13T^{2} \) |
| 17 | \( 1 + 0.968T + 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 23 | \( 1 + 5.21T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 + 0.0517T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 - 3.96T + 41T^{2} \) |
| 43 | \( 1 + 5.06T + 43T^{2} \) |
| 47 | \( 1 - 4.05T + 47T^{2} \) |
| 53 | \( 1 - 1.97T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 1.29T + 67T^{2} \) |
| 71 | \( 1 + 8.59T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 8.67T + 79T^{2} \) |
| 83 | \( 1 - 4.77T + 83T^{2} \) |
| 89 | \( 1 - 4.09T + 89T^{2} \) |
| 97 | \( 1 + 0.0180T + 97T^{2} \) |
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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
−7.86975615325059557979788947703, −7.43221250891918259910645446804, −6.71927443036135141980932195263, −5.68276471080845759238122750377, −4.84416382197467737847839931248, −4.21566886621655915969916677951, −3.65913645241451899043729728877, −2.52747143566848102095077280590, −1.75983981101741609632208248328, −0.46848598843894434694737368024,
0.46848598843894434694737368024, 1.75983981101741609632208248328, 2.52747143566848102095077280590, 3.65913645241451899043729728877, 4.21566886621655915969916677951, 4.84416382197467737847839931248, 5.68276471080845759238122750377, 6.71927443036135141980932195263, 7.43221250891918259910645446804, 7.86975615325059557979788947703