L(s) = 1 | − 3-s + 3·5-s − 7-s + 9-s + 5·11-s − 3·15-s − 6·19-s + 21-s + 2·23-s + 4·25-s − 27-s − 9·29-s + 3·31-s − 5·33-s − 3·35-s + 6·37-s − 6·41-s + 4·43-s + 3·45-s + 6·47-s + 49-s − 3·53-s + 15·55-s + 6·57-s + 13·59-s + 4·61-s − 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.50·11-s − 0.774·15-s − 1.37·19-s + 0.218·21-s + 0.417·23-s + 4/5·25-s − 0.192·27-s − 1.67·29-s + 0.538·31-s − 0.870·33-s − 0.507·35-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 0.447·45-s + 0.875·47-s + 1/7·49-s − 0.412·53-s + 2.02·55-s + 0.794·57-s + 1.69·59-s + 0.512·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.73133544052532, −12.36189165967891, −11.54940685239981, −11.47122322615912, −10.92887387157810, −10.24891063324725, −10.06103116636497, −9.584165446900076, −9.027202348607313, −8.880558918173784, −8.263563656752662, −7.472975915360837, −6.926399292499115, −6.717013855095925, −6.058438774469668, −5.854250130862974, −5.496944881129889, −4.651838532776674, −4.279547922232526, −3.803434765422880, −3.139130912152984, −2.408793166991948, −1.956794432172486, −1.428993482579253, −0.8533639611465525, 0,
0.8533639611465525, 1.428993482579253, 1.956794432172486, 2.408793166991948, 3.139130912152984, 3.803434765422880, 4.279547922232526, 4.651838532776674, 5.496944881129889, 5.854250130862974, 6.058438774469668, 6.717013855095925, 6.926399292499115, 7.472975915360837, 8.263563656752662, 8.880558918173784, 9.027202348607313, 9.584165446900076, 10.06103116636497, 10.24891063324725, 10.92887387157810, 11.47122322615912, 11.54940685239981, 12.36189165967891, 12.73133544052532