L(s) = 1 | − 0.815·2-s − 2.98·3-s − 1.33·4-s + 5-s + 2.43·6-s + 2.71·8-s + 5.92·9-s − 0.815·10-s + 2.42·11-s + 3.98·12-s + 5.44·13-s − 2.98·15-s + 0.454·16-s + 0.320·17-s − 4.82·18-s − 2.06·19-s − 1.33·20-s − 1.97·22-s + 23-s − 8.12·24-s + 25-s − 4.43·26-s − 8.72·27-s − 10.1·29-s + 2.43·30-s + 8.87·31-s − 5.80·32-s + ⋯ |
L(s) = 1 | − 0.576·2-s − 1.72·3-s − 0.667·4-s + 0.447·5-s + 0.993·6-s + 0.961·8-s + 1.97·9-s − 0.257·10-s + 0.730·11-s + 1.15·12-s + 1.51·13-s − 0.771·15-s + 0.113·16-s + 0.0776·17-s − 1.13·18-s − 0.473·19-s − 0.298·20-s − 0.420·22-s + 0.208·23-s − 1.65·24-s + 0.200·25-s − 0.870·26-s − 1.67·27-s − 1.87·29-s + 0.444·30-s + 1.59·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 0.815T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 - 0.320T + 17T^{2} \) |
| 19 | \( 1 + 2.06T + 19T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 + 8.66T + 37T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 4.60T + 59T^{2} \) |
| 61 | \( 1 + 3.63T + 61T^{2} \) |
| 67 | \( 1 + 1.54T + 67T^{2} \) |
| 71 | \( 1 - 6.66T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 1.08T + 83T^{2} \) |
| 89 | \( 1 + 4.65T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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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
−7.83378993873895593895767460061, −6.69122689599276835968324491912, −6.50843540425369993000738791129, −5.53625902444985307312237009541, −5.13680936557794034244924614292, −4.19589378052297358341261299586, −3.59838569465260715424517140029, −1.69561893872751446501997197585, −1.11270020986210389648666520987, 0,
1.11270020986210389648666520987, 1.69561893872751446501997197585, 3.59838569465260715424517140029, 4.19589378052297358341261299586, 5.13680936557794034244924614292, 5.53625902444985307312237009541, 6.50843540425369993000738791129, 6.69122689599276835968324491912, 7.83378993873895593895767460061