| L(s) = 1 | − 9-s − 6·19-s + 8·29-s + 14·31-s + 12·41-s + 13·49-s − 20·59-s + 2·61-s + 28·71-s + 16·79-s + 81-s + 12·101-s − 30·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 6·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.37·19-s + 1.48·29-s + 2.51·31-s + 1.87·41-s + 13/7·49-s − 2.60·59-s + 0.256·61-s + 3.32·71-s + 1.80·79-s + 1/9·81-s + 1.19·101-s − 2.87·109-s − 2·121-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 + 1.92·169-s + 0.458·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.081497910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.081497910\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 185 T^{2} + p^{2} T^{4} \) |
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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
−8.316135278026738864693456958040, −8.051961879923222673077970711170, −7.970949020447044529839592132840, −7.47092413132245305856124883170, −6.81714645204455009198314030039, −6.65672175128741560907409862380, −6.20814052585699346746704718872, −6.19636988814089754960505348976, −5.47128315572202408837661689058, −5.22480626763150479651552833548, −4.59940614615979281286986474824, −4.44048838516414432484857698573, −4.07195411272356769426395045363, −3.57878769141661893507591496728, −2.93993092537491935157029145286, −2.62487427599924820014779532100, −2.34094080351925932207094068322, −1.70142717203820727821630993050, −0.872304012292497807276476779638, −0.62336955883228869679325202408,
0.62336955883228869679325202408, 0.872304012292497807276476779638, 1.70142717203820727821630993050, 2.34094080351925932207094068322, 2.62487427599924820014779532100, 2.93993092537491935157029145286, 3.57878769141661893507591496728, 4.07195411272356769426395045363, 4.44048838516414432484857698573, 4.59940614615979281286986474824, 5.22480626763150479651552833548, 5.47128315572202408837661689058, 6.19636988814089754960505348976, 6.20814052585699346746704718872, 6.65672175128741560907409862380, 6.81714645204455009198314030039, 7.47092413132245305856124883170, 7.970949020447044529839592132840, 8.051961879923222673077970711170, 8.316135278026738864693456958040
