| L(s) = 1 | + 5-s + 7-s − 6·11-s − 4·13-s − 6·17-s + 2·19-s + 25-s − 6·29-s − 10·31-s + 35-s + 2·37-s + 6·41-s − 4·43-s + 49-s + 12·53-s − 6·55-s + 14·61-s − 4·65-s − 4·67-s − 6·71-s − 4·73-s − 6·77-s − 16·79-s + 12·83-s − 6·85-s − 6·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.80·11-s − 1.10·13-s − 1.45·17-s + 0.458·19-s + 1/5·25-s − 1.11·29-s − 1.79·31-s + 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 1.64·53-s − 0.809·55-s + 1.79·61-s − 0.496·65-s − 0.488·67-s − 0.712·71-s − 0.468·73-s − 0.683·77-s − 1.80·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.333092347145761586280629276704, −8.486796104743842391475144783781, −7.52901960668867754280433642901, −7.04398348805248716401259437199, −5.63223394326959203086574118577, −5.21354657812453653302539631933, −4.18698015044674762527792039200, −2.72394457334941735951674528638, −2.03122373506034402447940694337, 0,
2.03122373506034402447940694337, 2.72394457334941735951674528638, 4.18698015044674762527792039200, 5.21354657812453653302539631933, 5.63223394326959203086574118577, 7.04398348805248716401259437199, 7.52901960668867754280433642901, 8.486796104743842391475144783781, 9.333092347145761586280629276704
