| L(s) = 1 | + 5-s − 7-s − 2·11-s − 13-s + 3·17-s − 6·19-s − 4·23-s − 4·25-s − 2·29-s − 4·31-s − 35-s + 3·37-s + 5·43-s + 13·47-s − 6·49-s − 12·53-s − 2·55-s − 10·59-s − 8·61-s − 65-s + 2·67-s − 5·71-s − 10·73-s + 2·77-s + 4·79-s + 3·85-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s − 0.834·23-s − 4/5·25-s − 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.493·37-s + 0.762·43-s + 1.89·47-s − 6/7·49-s − 1.64·53-s − 0.269·55-s − 1.30·59-s − 1.02·61-s − 0.124·65-s + 0.244·67-s − 0.593·71-s − 1.17·73-s + 0.227·77-s + 0.450·79-s + 0.325·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−8.963928209525859216279788816156, −7.931818475756913627713932784539, −7.42085241738992224550819026024, −6.19695969612857050602479487947, −5.84229781714116440214520488100, −4.73920502052987077827176335423, −3.82010897415061793912288152862, −2.71625515373288996093576035666, −1.77354291627119599659363661895, 0,
1.77354291627119599659363661895, 2.71625515373288996093576035666, 3.82010897415061793912288152862, 4.73920502052987077827176335423, 5.84229781714116440214520488100, 6.19695969612857050602479487947, 7.42085241738992224550819026024, 7.931818475756913627713932784539, 8.963928209525859216279788816156
