| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.469750079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469750079\) |
| \(L(\frac{5}{2})\) |
|
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
−8.968483870366647127048734193145, −7.899492181163311703589294261157, −6.98322401260700648544472296795, −6.47295268971550636881682568625, −5.66722375234142070471035674916, −4.63630199511875558860635317980, −3.86447878562413846919079471243, −2.72034048943192975053245375747, −1.97725404593012011966472774398, −0.52544819686275328985980179722,
0.52544819686275328985980179722, 1.97725404593012011966472774398, 2.72034048943192975053245375747, 3.86447878562413846919079471243, 4.63630199511875558860635317980, 5.66722375234142070471035674916, 6.47295268971550636881682568625, 6.98322401260700648544472296795, 7.899492181163311703589294261157, 8.968483870366647127048734193145
