| L(s) = 1 | − 2·11-s − 2·13-s − 4·17-s + 19-s − 2·23-s − 5·25-s − 6·29-s + 4·31-s + 2·37-s + 2·41-s + 4·43-s − 10·47-s − 7·49-s − 6·53-s − 4·59-s + 2·61-s + 4·67-s − 4·71-s − 6·73-s + 8·79-s + 2·83-s − 6·89-s − 10·97-s − 4·101-s − 12·107-s − 2·109-s − 6·113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s − 0.417·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.328·37-s + 0.312·41-s + 0.609·43-s − 1.45·47-s − 49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.474·71-s − 0.702·73-s + 0.900·79-s + 0.219·83-s − 0.635·89-s − 1.01·97-s − 0.398·101-s − 1.16·107-s − 0.191·109-s − 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.368270182612817957769886359471, −8.237936218965864025222435287651, −7.68441424007857898366359376705, −6.73030131127953195720037644769, −5.86373516542821705353780780359, −4.94210606566121331850534027610, −4.07340272824547661174020500725, −2.88362293110800240687879207098, −1.85746581860673975293377108848, 0,
1.85746581860673975293377108848, 2.88362293110800240687879207098, 4.07340272824547661174020500725, 4.94210606566121331850534027610, 5.86373516542821705353780780359, 6.73030131127953195720037644769, 7.68441424007857898366359376705, 8.237936218965864025222435287651, 9.368270182612817957769886359471
