| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s + 2·7-s − 4·8-s − 4·10-s − 2·11-s − 4·14-s + 5·16-s + 6·20-s + 4·22-s − 4·23-s + 3·25-s + 6·28-s − 4·29-s − 6·32-s + 4·35-s + 8·37-s − 8·40-s + 4·41-s + 8·43-s − 6·44-s + 8·46-s + 3·49-s − 6·50-s − 12·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s − 1.41·8-s − 1.26·10-s − 0.603·11-s − 1.06·14-s + 5/4·16-s + 1.34·20-s + 0.852·22-s − 0.834·23-s + 3/5·25-s + 1.13·28-s − 0.742·29-s − 1.06·32-s + 0.676·35-s + 1.31·37-s − 1.26·40-s + 0.624·41-s + 1.21·43-s − 0.904·44-s + 1.17·46-s + 3/7·49-s − 0.848·50-s − 1.64·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.179874039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.179874039\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 94 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.069010067391974921787754007440, −7.86817091426692951286676760735, −7.45528631651160523902631569633, −7.44159966918481628693241872822, −6.67645728759025185010571930130, −6.49695804129831592607330074785, −6.02978089617071476213042676222, −5.90624719315265733568864390077, −5.24807301524678661547440328364, −5.23563599171944562561386112902, −4.42277320494169648085506626055, −4.40666001660186538206808704401, −3.48121124671819618694571974185, −3.37816480894177620080557535980, −2.61843441706373356850271154758, −2.26015750238241324990905258330, −2.02277669445611681829690769554, −1.60998436757552426509478931467, −0.829829878682742375016200558201, −0.58675343213014481012918381257,
0.58675343213014481012918381257, 0.829829878682742375016200558201, 1.60998436757552426509478931467, 2.02277669445611681829690769554, 2.26015750238241324990905258330, 2.61843441706373356850271154758, 3.37816480894177620080557535980, 3.48121124671819618694571974185, 4.40666001660186538206808704401, 4.42277320494169648085506626055, 5.23563599171944562561386112902, 5.24807301524678661547440328364, 5.90624719315265733568864390077, 6.02978089617071476213042676222, 6.49695804129831592607330074785, 6.67645728759025185010571930130, 7.44159966918481628693241872822, 7.45528631651160523902631569633, 7.86817091426692951286676760735, 8.069010067391974921787754007440
