| L(s) = 1 | + 3-s − 4·5-s + 2·7-s + 9-s − 4·11-s − 4·15-s + 2·17-s − 2·19-s + 2·21-s + 11·25-s + 27-s + 6·29-s + 10·31-s − 4·33-s − 8·35-s + 10·37-s − 8·41-s − 4·43-s − 4·45-s + 4·47-s − 3·49-s + 2·51-s + 10·53-s + 16·55-s − 2·57-s − 8·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 11/5·25-s + 0.192·27-s + 1.11·29-s + 1.79·31-s − 0.696·33-s − 1.35·35-s + 1.64·37-s − 1.24·41-s − 0.609·43-s − 0.596·45-s + 0.583·47-s − 3/7·49-s + 0.280·51-s + 1.37·53-s + 2.15·55-s − 0.264·57-s − 1.04·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780044325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780044325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.14329800111145, −14.69770414869234, −14.07346199507467, −13.42676675693798, −12.96751235400227, −12.27371186423823, −11.84813988339899, −11.45059524921293, −10.79674254015996, −10.26856218020200, −9.818014735368456, −8.752047366240232, −8.251117590788308, −8.174563530063013, −7.577909374909223, −7.034569580703220, −6.348562803912572, −5.363332422125680, −4.725642524130762, −4.359028791004699, −3.703745966498387, −2.867807910896861, −2.568624189220175, −1.325927307467221, −0.5233348011855495,
0.5233348011855495, 1.325927307467221, 2.568624189220175, 2.867807910896861, 3.703745966498387, 4.359028791004699, 4.725642524130762, 5.363332422125680, 6.348562803912572, 7.034569580703220, 7.577909374909223, 8.174563530063013, 8.251117590788308, 8.752047366240232, 9.818014735368456, 10.26856218020200, 10.79674254015996, 11.45059524921293, 11.84813988339899, 12.27371186423823, 12.96751235400227, 13.42676675693798, 14.07346199507467, 14.69770414869234, 15.14329800111145
