L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 4·7-s − 8-s + 9-s + 6·11-s + 2·12-s + 2·13-s − 4·14-s + 16-s − 17-s − 18-s + 4·19-s + 8·21-s − 6·22-s − 2·24-s − 5·25-s − 2·26-s − 4·27-s + 4·28-s − 4·31-s − 32-s + 12·33-s + 34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.577·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s + 1.74·21-s − 1.27·22-s − 0.408·24-s − 25-s − 0.392·26-s − 0.769·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s + 2.08·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62866 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62866 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.548544497\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.548544497\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.38099055872564, −13.95153879813623, −13.49610086513020, −12.77015526921224, −11.99065817549700, −11.60018832379888, −11.26677615198893, −10.82403690478986, −9.946331754779960, −9.409904639396544, −9.085546152463763, −8.697543948492074, −7.954682111809038, −7.864679825114635, −7.215651319664046, −6.520068432191183, −5.927298940517305, −5.284078204759776, −4.452148873979697, −3.857193642760091, −3.477507202233385, −2.565091099784354, −1.896430621063719, −1.483891290427476, −0.7941619781895600,
0.7941619781895600, 1.483891290427476, 1.896430621063719, 2.565091099784354, 3.477507202233385, 3.857193642760091, 4.452148873979697, 5.284078204759776, 5.927298940517305, 6.520068432191183, 7.215651319664046, 7.864679825114635, 7.954682111809038, 8.697543948492074, 9.085546152463763, 9.409904639396544, 9.946331754779960, 10.82403690478986, 11.26677615198893, 11.60018832379888, 11.99065817549700, 12.77015526921224, 13.49610086513020, 13.95153879813623, 14.38099055872564