| L(s) = 1 | + 3-s − 2·4-s − 5-s − 2·7-s − 2·9-s + 2·11-s − 2·12-s − 13-s − 15-s + 4·16-s + 2·17-s + 4·19-s + 2·20-s − 2·21-s − 2·23-s + 25-s − 5·27-s + 4·28-s + 10·29-s + 4·31-s + 2·33-s + 2·35-s + 4·36-s − 10·37-s − 39-s + 10·41-s + 11·43-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.603·11-s − 0.577·12-s − 0.277·13-s − 0.258·15-s + 16-s + 0.485·17-s + 0.917·19-s + 0.447·20-s − 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.962·27-s + 0.755·28-s + 1.85·29-s + 0.718·31-s + 0.348·33-s + 0.338·35-s + 2/3·36-s − 1.64·37-s − 0.160·39-s + 1.56·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.00364147496623, −13.81400318347423, −12.93330563657238, −12.63908374415745, −12.02218589641380, −11.81939811134803, −11.04974510313397, −10.36236454519924, −9.974358390748584, −9.409121226126519, −9.061953256739655, −8.635955763377290, −7.991349414753073, −7.720232898538469, −7.038689138270668, −6.299910503337535, −5.906358333475932, −5.230780586306507, −4.661190886884037, −4.051902360590314, −3.577930239870910, −2.944792395378622, −2.676106360282631, −1.449267388375754, −0.7848588040937814, 0,
0.7848588040937814, 1.449267388375754, 2.676106360282631, 2.944792395378622, 3.577930239870910, 4.051902360590314, 4.661190886884037, 5.230780586306507, 5.906358333475932, 6.299910503337535, 7.038689138270668, 7.720232898538469, 7.991349414753073, 8.635955763377290, 9.061953256739655, 9.409121226126519, 9.974358390748584, 10.36236454519924, 11.04974510313397, 11.81939811134803, 12.02218589641380, 12.63908374415745, 12.93330563657238, 13.81400318347423, 14.00364147496623
