| L(s) = 1 | + 5·5-s + 17·7-s − 30·11-s − 61·13-s − 120·17-s − 43·19-s + 90·23-s + 25·25-s − 90·29-s + 8·31-s + 85·35-s + 317·37-s − 30·41-s − 220·43-s − 180·47-s − 54·49-s − 630·53-s − 150·55-s − 840·59-s + 599·61-s − 305·65-s + 107·67-s − 210·71-s − 421·73-s − 510·77-s + 353·79-s − 1.35e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.917·7-s − 0.822·11-s − 1.30·13-s − 1.71·17-s − 0.519·19-s + 0.815·23-s + 1/5·25-s − 0.576·29-s + 0.0463·31-s + 0.410·35-s + 1.40·37-s − 0.114·41-s − 0.780·43-s − 0.558·47-s − 0.157·49-s − 1.63·53-s − 0.367·55-s − 1.85·59-s + 1.25·61-s − 0.582·65-s + 0.195·67-s − 0.351·71-s − 0.674·73-s − 0.754·77-s + 0.502·79-s − 1.78·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 7 | \( 1 - 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 61 T + p^{3} T^{2} \) |
| 17 | \( 1 + 120 T + p^{3} T^{2} \) |
| 19 | \( 1 + 43 T + p^{3} T^{2} \) |
| 23 | \( 1 - 90 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 317 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 + 220 T + p^{3} T^{2} \) |
| 47 | \( 1 + 180 T + p^{3} T^{2} \) |
| 53 | \( 1 + 630 T + p^{3} T^{2} \) |
| 59 | \( 1 + 840 T + p^{3} T^{2} \) |
| 61 | \( 1 - 599 T + p^{3} T^{2} \) |
| 67 | \( 1 - 107 T + p^{3} T^{2} \) |
| 71 | \( 1 + 210 T + p^{3} T^{2} \) |
| 73 | \( 1 + 421 T + p^{3} T^{2} \) |
| 79 | \( 1 - 353 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1350 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 + 997 T + p^{3} T^{2} \) |
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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
−9.973472488516440348843181993228, −9.104516833663772132678806120767, −8.165224098107669032733588206007, −7.31375219846346295337085926424, −6.30448159999742980993844178670, −5.04377952131409307916221532847, −4.54463304548614176736361603100, −2.74643821316646438203047298447, −1.83649010473963576289456119762, 0,
1.83649010473963576289456119762, 2.74643821316646438203047298447, 4.54463304548614176736361603100, 5.04377952131409307916221532847, 6.30448159999742980993844178670, 7.31375219846346295337085926424, 8.165224098107669032733588206007, 9.104516833663772132678806120767, 9.973472488516440348843181993228
