| L(s) = 1 | + 7-s + 2·11-s − 3·13-s + 23-s − 5·25-s − 29-s − 5·31-s − 8·37-s + 7·41-s − 4·43-s − 3·47-s + 49-s + 12·53-s − 4·59-s − 6·61-s − 12·67-s − 13·71-s + 3·73-s + 2·77-s + 4·79-s − 16·83-s − 4·89-s − 3·91-s + 10·97-s + 10·101-s + 14·103-s + 8·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.603·11-s − 0.832·13-s + 0.208·23-s − 25-s − 0.185·29-s − 0.898·31-s − 1.31·37-s + 1.09·41-s − 0.609·43-s − 0.437·47-s + 1/7·49-s + 1.64·53-s − 0.520·59-s − 0.768·61-s − 1.46·67-s − 1.54·71-s + 0.351·73-s + 0.227·77-s + 0.450·79-s − 1.75·83-s − 0.423·89-s − 0.314·91-s + 1.01·97-s + 0.995·101-s + 1.37·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.47650185401447785350056395557, −7.33373344127595624040982800606, −6.28523960056601843933881102851, −5.61024586075110867468737675807, −4.83581985298052415442118959104, −4.10039885697875862994496279724, −3.28744257614824520184549874744, −2.24773642619845547091975353441, −1.43071464639639147307854140846, 0,
1.43071464639639147307854140846, 2.24773642619845547091975353441, 3.28744257614824520184549874744, 4.10039885697875862994496279724, 4.83581985298052415442118959104, 5.61024586075110867468737675807, 6.28523960056601843933881102851, 7.33373344127595624040982800606, 7.47650185401447785350056395557
