L(s) = 1 | + 5-s + 5·11-s − 3·13-s − 17-s − 6·19-s + 6·23-s + 25-s − 9·29-s − 4·31-s − 2·37-s − 4·41-s + 10·43-s + 47-s + 4·53-s + 5·55-s − 8·59-s − 8·61-s − 3·65-s + 12·67-s + 8·71-s − 2·73-s − 13·79-s − 4·83-s − 85-s + 4·89-s − 6·95-s + 13·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.50·11-s − 0.832·13-s − 0.242·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s − 0.328·37-s − 0.624·41-s + 1.52·43-s + 0.145·47-s + 0.549·53-s + 0.674·55-s − 1.04·59-s − 1.02·61-s − 0.372·65-s + 1.46·67-s + 0.949·71-s − 0.234·73-s − 1.46·79-s − 0.439·83-s − 0.108·85-s + 0.423·89-s − 0.615·95-s + 1.31·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.66957798076799, −13.01447271608588, −12.74772900920421, −12.30918398996705, −11.70838604206371, −11.17639573672303, −10.85552103745283, −10.32078170487504, −9.605910250364563, −9.310514600214052, −8.872726190978409, −8.522172381011433, −7.602827699260639, −7.220617120663850, −6.787643467715351, −6.200033311596637, −5.797979769055572, −5.105075800065362, −4.601116393181200, −3.998613458242602, −3.559345693895388, −2.791394622334571, −2.075933616297545, −1.708835310232804, −0.8778340055990263, 0,
0.8778340055990263, 1.708835310232804, 2.075933616297545, 2.791394622334571, 3.559345693895388, 3.998613458242602, 4.601116393181200, 5.105075800065362, 5.797979769055572, 6.200033311596637, 6.787643467715351, 7.220617120663850, 7.602827699260639, 8.522172381011433, 8.872726190978409, 9.310514600214052, 9.605910250364563, 10.32078170487504, 10.85552103745283, 11.17639573672303, 11.70838604206371, 12.30918398996705, 12.74772900920421, 13.01447271608588, 13.66957798076799