| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 7-s + 4·8-s − 4·10-s + 2·11-s + 4·13-s + 2·14-s + 5·16-s + 3·17-s + 7·19-s − 6·20-s + 4·22-s + 6·23-s + 3·25-s + 8·26-s + 3·28-s + 3·29-s + 31-s + 6·32-s + 6·34-s − 2·35-s + 13·37-s + 14·38-s − 8·40-s − 8·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.377·7-s + 1.41·8-s − 1.26·10-s + 0.603·11-s + 1.10·13-s + 0.534·14-s + 5/4·16-s + 0.727·17-s + 1.60·19-s − 1.34·20-s + 0.852·22-s + 1.25·23-s + 3/5·25-s + 1.56·26-s + 0.566·28-s + 0.557·29-s + 0.179·31-s + 1.06·32-s + 1.02·34-s − 0.338·35-s + 2.13·37-s + 2.27·38-s − 1.26·40-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.289079855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.289079855\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13 T + 108 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 172 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−10.18117660127891089939227519679, −9.987167497693696907981490082249, −9.274576879196523008943501056100, −9.033087181020552502331228861761, −8.215677478566040724031925613344, −8.122791105403659904684444900838, −7.55886488555460606639278772860, −7.24890440236121834873346396090, −6.59502373629883656343250751602, −6.45968307937970757559734997664, −5.59914590068054735675065520320, −5.58519302697905327488577221262, −4.75387261308194036035165751564, −4.54878270126902218209876285934, −3.98407204612482346710023372520, −3.42710888140596097956805943737, −3.10831468308433265340113329022, −2.63434882131933127885518227886, −1.34411130567678245509630805146, −1.16615174006350466134066828480,
1.16615174006350466134066828480, 1.34411130567678245509630805146, 2.63434882131933127885518227886, 3.10831468308433265340113329022, 3.42710888140596097956805943737, 3.98407204612482346710023372520, 4.54878270126902218209876285934, 4.75387261308194036035165751564, 5.58519302697905327488577221262, 5.59914590068054735675065520320, 6.45968307937970757559734997664, 6.59502373629883656343250751602, 7.24890440236121834873346396090, 7.55886488555460606639278772860, 8.122791105403659904684444900838, 8.215677478566040724031925613344, 9.033087181020552502331228861761, 9.274576879196523008943501056100, 9.987167497693696907981490082249, 10.18117660127891089939227519679
