| L(s) = 1 | + 3-s − i·5-s − i·7-s + 9-s + i·11-s + (3 − 2i)13-s − i·15-s + 17-s − i·19-s − i·21-s − 3·23-s + 4·25-s + 27-s + 9·29-s − 4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447i·5-s − 0.377i·7-s + 0.333·9-s + 0.301i·11-s + (0.832 − 0.554i)13-s − 0.258i·15-s + 0.242·17-s − 0.229i·19-s − 0.218i·21-s − 0.625·23-s + 0.800·25-s + 0.192·27-s + 1.67·29-s − 0.718i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.533948519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.533948519\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 + iT - 5T^{2} \) |
| 11 | \( 1 - iT - 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.210465092373756938395405768065, −7.79115696536499173097832906244, −6.74525447372785304390352163120, −6.22463069969573451966275800619, −5.09502246810077973547265163399, −4.53059312845347709678710627573, −3.58590438599604516299738259609, −2.88792131916158959576333128005, −1.72469667807931057524271512082, −0.74565505245703246239268931646,
1.14420539932075394704487982025, 2.21786777961206874788134978040, 3.08248313271585598813811692921, 3.76023782832021069961392555178, 4.67584599395766886895562553605, 5.58731175634270705108304064385, 6.48424180932012761296368316578, 6.88422382703698985366006553210, 7.950802183865644752565503791368, 8.480269988502462033873270785455
