L(s) = 1 | + 6·9-s − 8·11-s − 12·29-s − 16·31-s + 4·41-s − 49-s − 16·59-s − 28·61-s + 32·71-s − 16·79-s + 27·81-s − 20·89-s − 48·99-s − 12·101-s − 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + ⋯ |
L(s) = 1 | + 2·9-s − 2.41·11-s − 2.22·29-s − 2.87·31-s + 0.624·41-s − 1/7·49-s − 2.08·59-s − 3.58·61-s + 3.79·71-s − 1.80·79-s + 3·81-s − 2.11·89-s − 4.82·99-s − 1.19·101-s − 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5572473402\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5572473402\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.295547717678141579623653251891, −8.377609988003526667801267576654, −8.265899950052732115419759938554, −7.63705964918998725770461433039, −7.45560266832340850254893034282, −7.24858340210446861545299137651, −7.04534924918139999345931367787, −6.11208397826544489742084323069, −6.01725356963673534933512591456, −5.29690891153501965584474231139, −5.27684196099387196231529700968, −4.67549331470859371004058129327, −4.35358457679376603801551612536, −3.68288765754575919343428686690, −3.55905166563266995484188735299, −2.80724664657204643598495483787, −2.33132608666228447039110626194, −1.64769041325788342416736408368, −1.55284920240425145576299385101, −0.22725873497587523262646158805,
0.22725873497587523262646158805, 1.55284920240425145576299385101, 1.64769041325788342416736408368, 2.33132608666228447039110626194, 2.80724664657204643598495483787, 3.55905166563266995484188735299, 3.68288765754575919343428686690, 4.35358457679376603801551612536, 4.67549331470859371004058129327, 5.27684196099387196231529700968, 5.29690891153501965584474231139, 6.01725356963673534933512591456, 6.11208397826544489742084323069, 7.04534924918139999345931367787, 7.24858340210446861545299137651, 7.45560266832340850254893034282, 7.63705964918998725770461433039, 8.265899950052732115419759938554, 8.377609988003526667801267576654, 9.295547717678141579623653251891