| L(s) = 1 | + 5-s − 7-s − 3·11-s − 4·13-s + 3·17-s + 19-s + 8·23-s − 4·25-s + 2·31-s − 35-s − 8·37-s + 11·43-s + 7·47-s − 6·49-s − 2·53-s − 3·55-s − 6·59-s − 61-s − 4·65-s − 10·67-s − 2·71-s + 5·73-s + 3·77-s − 2·79-s + 3·85-s − 6·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.904·11-s − 1.10·13-s + 0.727·17-s + 0.229·19-s + 1.66·23-s − 4/5·25-s + 0.359·31-s − 0.169·35-s − 1.31·37-s + 1.67·43-s + 1.02·47-s − 6/7·49-s − 0.274·53-s − 0.404·55-s − 0.781·59-s − 0.128·61-s − 0.496·65-s − 1.22·67-s − 0.237·71-s + 0.585·73-s + 0.341·77-s − 0.225·79-s + 0.325·85-s − 0.635·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.62930955652200617425472221580, −7.24496051993742145141374877693, −6.37053634668083859118775479546, −5.46731353162142922774499464367, −5.12472176266628188623566855372, −4.13055323242240284411324928497, −3.01902814209356883750354121381, −2.55684717596332807065167356892, −1.35184324636083432392622951369, 0,
1.35184324636083432392622951369, 2.55684717596332807065167356892, 3.01902814209356883750354121381, 4.13055323242240284411324928497, 5.12472176266628188623566855372, 5.46731353162142922774499464367, 6.37053634668083859118775479546, 7.24496051993742145141374877693, 7.62930955652200617425472221580
