L(s) = 1 | − 5-s − 4·7-s + 4·11-s + 2·13-s − 6·17-s + 6·19-s − 6·23-s + 25-s + 2·29-s + 4·31-s + 4·35-s + 8·37-s − 8·41-s + 43-s − 6·47-s + 9·49-s − 6·53-s − 4·55-s − 10·61-s − 2·65-s − 12·67-s + 16·71-s + 16·73-s − 16·77-s + 4·79-s + 6·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 1.37·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s + 1.31·37-s − 1.24·41-s + 0.152·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 0.539·55-s − 1.28·61-s − 0.248·65-s − 1.46·67-s + 1.89·71-s + 1.87·73-s − 1.82·77-s + 0.450·79-s + 0.658·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.48849155552248, −15.00617997735590, −14.12304068803770, −13.72673538404898, −13.37482013578134, −12.66544558071299, −12.13793945461073, −11.72027893520235, −11.19571426341070, −10.54032288676623, −9.842558821266034, −9.348979524811985, −9.138974560439199, −8.191454702292971, −7.849837511010118, −6.798006081941379, −6.575282281765516, −6.207540254966218, −5.351162225792665, −4.446324637975131, −3.981556383801455, −3.346675728140388, −2.834437690352454, −1.829447984831378, −0.9044209494024330, 0,
0.9044209494024330, 1.829447984831378, 2.834437690352454, 3.346675728140388, 3.981556383801455, 4.446324637975131, 5.351162225792665, 6.207540254966218, 6.575282281765516, 6.798006081941379, 7.849837511010118, 8.191454702292971, 9.138974560439199, 9.348979524811985, 9.842558821266034, 10.54032288676623, 11.19571426341070, 11.72027893520235, 12.13793945461073, 12.66544558071299, 13.37482013578134, 13.72673538404898, 14.12304068803770, 15.00617997735590, 15.48849155552248