| L(s) = 1 | + 3·7-s + 3·11-s + 13-s + 17-s − 4·19-s + 2·23-s − 7·29-s − 5·31-s + 6·37-s − 4·41-s − 6·43-s − 13·47-s + 2·49-s − 9·53-s + 5·59-s + 13·61-s + 5·67-s − 2·73-s + 9·77-s + 14·79-s − 9·83-s − 4·89-s + 3·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s + 0.277·13-s + 0.242·17-s − 0.917·19-s + 0.417·23-s − 1.29·29-s − 0.898·31-s + 0.986·37-s − 0.624·41-s − 0.914·43-s − 1.89·47-s + 2/7·49-s − 1.23·53-s + 0.650·59-s + 1.66·61-s + 0.610·67-s − 0.234·73-s + 1.02·77-s + 1.57·79-s − 0.987·83-s − 0.423·89-s + 0.314·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.80977229043896, −14.51276728606195, −13.95305555525641, −13.19076812059701, −12.90601517452836, −12.29636669991698, −11.52970283997014, −11.21261339146630, −11.04876738157490, −10.08467168977684, −9.663152383333550, −9.059393242893991, −8.398747551002777, −8.174731530272043, −7.432593361729380, −6.847481587735919, −6.320560538474247, −5.658807944412351, −4.996558191748518, −4.575279091676304, −3.754285139188126, −3.417459465383991, −2.287197971625739, −1.735263851320637, −1.155667663669641, 0,
1.155667663669641, 1.735263851320637, 2.287197971625739, 3.417459465383991, 3.754285139188126, 4.575279091676304, 4.996558191748518, 5.658807944412351, 6.320560538474247, 6.847481587735919, 7.432593361729380, 8.174731530272043, 8.398747551002777, 9.059393242893991, 9.663152383333550, 10.08467168977684, 11.04876738157490, 11.21261339146630, 11.52970283997014, 12.29636669991698, 12.90601517452836, 13.19076812059701, 13.95305555525641, 14.51276728606195, 14.80977229043896
