L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s − 4·13-s + 4·15-s − 25-s − 4·27-s + 20·37-s + 8·39-s − 4·41-s − 24·43-s − 6·45-s + 10·49-s + 20·53-s + 8·65-s − 16·67-s + 8·71-s + 2·75-s + 16·79-s + 5·81-s − 8·83-s + 12·89-s − 8·107-s − 40·111-s − 12·117-s + 18·121-s + 8·123-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s + 1.03·15-s − 1/5·25-s − 0.769·27-s + 3.28·37-s + 1.28·39-s − 0.624·41-s − 3.65·43-s − 0.894·45-s + 10/7·49-s + 2.74·53-s + 0.992·65-s − 1.95·67-s + 0.949·71-s + 0.230·75-s + 1.80·79-s + 5/9·81-s − 0.878·83-s + 1.27·89-s − 0.773·107-s − 3.79·111-s − 1.10·117-s + 1.63·121-s + 0.721·123-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
\(L(1)\) |
\(\approx\) |
\(0.6787482142\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6787482142\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−8.740322834163485022205936096313, −8.102577254143068176026485834363, −7.934118290087340528483643131443, −7.38839189272073119099081381015, −7.36755316879797070452781670454, −6.72796400532654923666995763302, −6.57466678806329785547386722874, −6.03727900008917443602226944378, −5.71531446150005638769356676506, −5.26283856781323370648246448306, −4.90353287781904129128496325244, −4.52287740945337835369967547220, −4.27085294422757638879633628040, −3.56232073015672390161557220473, −3.52486949339636624436067475171, −2.49586402847518774850270032372, −2.43855041718298952145646574130, −1.58670216669201233258972177573, −0.915494956748907366418585187790, −0.32908894437133172800253090557,
0.32908894437133172800253090557, 0.915494956748907366418585187790, 1.58670216669201233258972177573, 2.43855041718298952145646574130, 2.49586402847518774850270032372, 3.52486949339636624436067475171, 3.56232073015672390161557220473, 4.27085294422757638879633628040, 4.52287740945337835369967547220, 4.90353287781904129128496325244, 5.26283856781323370648246448306, 5.71531446150005638769356676506, 6.03727900008917443602226944378, 6.57466678806329785547386722874, 6.72796400532654923666995763302, 7.36755316879797070452781670454, 7.38839189272073119099081381015, 7.934118290087340528483643131443, 8.102577254143068176026485834363, 8.740322834163485022205936096313