| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 11-s − 4·13-s − 2·14-s + 16-s − 6·19-s + 20-s + 22-s − 2·23-s + 25-s + 4·26-s + 2·28-s − 6·29-s + 4·31-s − 32-s + 2·35-s + 2·37-s + 6·38-s − 40-s − 6·41-s + 6·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 1.37·19-s + 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.338·35-s + 0.328·37-s + 0.973·38-s − 0.158·40-s − 0.937·41-s + 0.914·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.78484165128935, −12.42815682033103, −12.08883622987728, −11.46410374322458, −10.98532531670509, −10.66294225159018, −10.21963829266236, −9.638754540166731, −9.413651810393083, −8.702974802487980, −8.407977036282577, −7.831070785352596, −7.503248595879231, −6.935159757605260, −6.486118491454295, −5.804054502379378, −5.532866127112191, −4.805749577666999, −4.408349953152583, −3.850703436654980, −2.975095858774933, −2.468052155359020, −2.009851151011876, −1.564941844646865, −0.6859697489045103, 0,
0.6859697489045103, 1.564941844646865, 2.009851151011876, 2.468052155359020, 2.975095858774933, 3.850703436654980, 4.408349953152583, 4.805749577666999, 5.532866127112191, 5.804054502379378, 6.486118491454295, 6.935159757605260, 7.503248595879231, 7.831070785352596, 8.407977036282577, 8.702974802487980, 9.413651810393083, 9.638754540166731, 10.21963829266236, 10.66294225159018, 10.98532531670509, 11.46410374322458, 12.08883622987728, 12.42815682033103, 12.78484165128935
