| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s + 2·15-s + 2·17-s + 4·19-s + 8·23-s − 25-s − 27-s − 6·29-s + 8·31-s + 4·33-s + 6·37-s + 6·41-s + 4·43-s − 2·45-s − 7·49-s − 2·51-s + 2·53-s + 8·55-s − 4·57-s − 4·59-s + 2·61-s + 4·67-s − 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 49-s − 0.280·51-s + 0.274·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.963·69-s + 0.949·71-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
| \(L(1)\) |
\(\approx\) |
\(1.322411377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322411377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.29724008168984, −14.63159491316374, −13.96760997875372, −13.24696342935446, −12.97092397406846, −12.34901920382623, −11.81574526546157, −11.25862791255330, −10.97001985117279, −10.33658379158793, −9.657556378976405, −9.263721744558210, −8.362456785462266, −7.787064428132640, −7.536385184030891, −6.903995065676531, −6.105276261704502, −5.530088011963957, −4.952123668809046, −4.467394274054526, −3.629557292718827, −3.035148318770331, −2.355034654515254, −1.155738794431821, −0.5202191277635564,
0.5202191277635564, 1.155738794431821, 2.355034654515254, 3.035148318770331, 3.629557292718827, 4.467394274054526, 4.952123668809046, 5.530088011963957, 6.105276261704502, 6.903995065676531, 7.536385184030891, 7.787064428132640, 8.362456785462266, 9.263721744558210, 9.657556378976405, 10.33658379158793, 10.97001985117279, 11.25862791255330, 11.81574526546157, 12.34901920382623, 12.97092397406846, 13.24696342935446, 13.96760997875372, 14.63159491316374, 15.29724008168984
