| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 8·13-s + 8·15-s + 11·25-s + 4·27-s − 16·31-s − 8·37-s + 16·39-s − 12·41-s − 8·43-s + 12·45-s − 2·49-s + 24·53-s + 32·65-s − 24·67-s + 32·71-s + 22·75-s + 16·79-s + 5·81-s − 24·83-s − 20·89-s − 32·93-s + 24·107-s − 16·111-s + 24·117-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 2.21·13-s + 2.06·15-s + 11/5·25-s + 0.769·27-s − 2.87·31-s − 1.31·37-s + 2.56·39-s − 1.87·41-s − 1.21·43-s + 1.78·45-s − 2/7·49-s + 3.29·53-s + 3.96·65-s − 2.93·67-s + 3.79·71-s + 2.54·75-s + 1.80·79-s + 5/9·81-s − 2.63·83-s − 2.11·89-s − 3.31·93-s + 2.32·107-s − 1.51·111-s + 2.21·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.858966141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.858966141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.593103862997289515161149978480, −8.465074119072821471934727360706, −8.313713077971899769957175927030, −7.42711791949088408360023090515, −7.16864354586200005252454267837, −6.93349447752719607572669495088, −6.35465224415703681644066911794, −6.22819068044509243053407800261, −5.57973845165202652851302566965, −5.32726177318077133538122494798, −5.20228703611554306028487723627, −4.38196777028795650468997444597, −3.80999017682345110126137479050, −3.66443547209894230412324752838, −3.17207410894251954689834562891, −2.80934343359658807365406388520, −1.94019893430483449593711057771, −1.81576533597752949496524384174, −1.58407720160088822587204329960, −0.72935541335976507216671075768,
0.72935541335976507216671075768, 1.58407720160088822587204329960, 1.81576533597752949496524384174, 1.94019893430483449593711057771, 2.80934343359658807365406388520, 3.17207410894251954689834562891, 3.66443547209894230412324752838, 3.80999017682345110126137479050, 4.38196777028795650468997444597, 5.20228703611554306028487723627, 5.32726177318077133538122494798, 5.57973845165202652851302566965, 6.22819068044509243053407800261, 6.35465224415703681644066911794, 6.93349447752719607572669495088, 7.16864354586200005252454267837, 7.42711791949088408360023090515, 8.313713077971899769957175927030, 8.465074119072821471934727360706, 8.593103862997289515161149978480
