| L(s) = 1 | + 2·3-s + 2·7-s + 3·9-s + 4·11-s + 8·17-s − 2·19-s + 4·21-s + 4·23-s + 2·25-s + 4·27-s − 12·29-s + 8·33-s + 20·37-s − 4·41-s − 8·47-s + 3·49-s + 16·51-s + 4·53-s − 4·57-s − 8·59-s + 4·61-s + 6·63-s − 12·67-s + 8·69-s + 8·71-s + 20·73-s + 4·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.755·7-s + 9-s + 1.20·11-s + 1.94·17-s − 0.458·19-s + 0.872·21-s + 0.834·23-s + 2/5·25-s + 0.769·27-s − 2.22·29-s + 1.39·33-s + 3.28·37-s − 0.624·41-s − 1.16·47-s + 3/7·49-s + 2.24·51-s + 0.549·53-s − 0.529·57-s − 1.04·59-s + 0.512·61-s + 0.755·63-s − 1.46·67-s + 0.963·69-s + 0.949·71-s + 2.34·73-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40755456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.354267279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.354267279\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 246 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.237127430986987223703497978677, −7.77123754008105089067806523473, −7.56633354999644910136267475283, −7.36200330013587989465303721764, −6.94109368742087712520403689092, −6.46975516833232188526753029814, −5.94763026386336898376728060579, −5.93086990862838705110021157912, −5.27722832395230481767040893059, −4.93040176990576066936930357227, −4.44330866103192909201071991254, −4.26178679478939778542792197435, −3.58625787985966202761457114244, −3.50959091310682604742774821612, −3.07523923905604799118726174833, −2.56110164366120544260213575225, −1.95065131414752227640536805844, −1.74134081642052532664763440021, −1.06998867222435167412970581973, −0.76980208862776593223650390243,
0.76980208862776593223650390243, 1.06998867222435167412970581973, 1.74134081642052532664763440021, 1.95065131414752227640536805844, 2.56110164366120544260213575225, 3.07523923905604799118726174833, 3.50959091310682604742774821612, 3.58625787985966202761457114244, 4.26178679478939778542792197435, 4.44330866103192909201071991254, 4.93040176990576066936930357227, 5.27722832395230481767040893059, 5.93086990862838705110021157912, 5.94763026386336898376728060579, 6.46975516833232188526753029814, 6.94109368742087712520403689092, 7.36200330013587989465303721764, 7.56633354999644910136267475283, 7.77123754008105089067806523473, 8.237127430986987223703497978677
