| L(s) = 1 | − 3·5-s + 4·7-s + 13-s + 3·17-s − 4·19-s + 4·25-s + 9·29-s + 4·31-s − 12·35-s + 37-s − 6·41-s + 8·43-s − 12·47-s + 9·49-s − 6·53-s + 61-s − 3·65-s − 4·67-s − 12·71-s + 11·73-s + 16·79-s + 12·83-s − 9·85-s + 3·89-s + 4·91-s + 12·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 1.51·7-s + 0.277·13-s + 0.727·17-s − 0.917·19-s + 4/5·25-s + 1.67·29-s + 0.718·31-s − 2.02·35-s + 0.164·37-s − 0.937·41-s + 1.21·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s + 0.128·61-s − 0.372·65-s − 0.488·67-s − 1.42·71-s + 1.28·73-s + 1.80·79-s + 1.31·83-s − 0.976·85-s + 0.317·89-s + 0.419·91-s + 1.23·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.794431993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.794431993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.153205859443475808039169365129, −7.79409315011386614496311102735, −6.88462097851938764634300699739, −6.11563543838000124898248269154, −4.97067099118845305707332827681, −4.60568685013129480432704821683, −3.83528584973290190499044501865, −2.94903686287780602993956585385, −1.78014361920684476225734672011, −0.75418881749976335996165607688,
0.75418881749976335996165607688, 1.78014361920684476225734672011, 2.94903686287780602993956585385, 3.83528584973290190499044501865, 4.60568685013129480432704821683, 4.97067099118845305707332827681, 6.11563543838000124898248269154, 6.88462097851938764634300699739, 7.79409315011386614496311102735, 8.153205859443475808039169365129
