L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s + 3·9-s − 4·12-s − 4·16-s − 6·18-s − 4·27-s + 20·31-s + 8·32-s + 6·36-s + 8·37-s + 20·41-s + 8·43-s + 8·48-s + 10·49-s + 20·53-s + 8·54-s − 40·62-s − 8·64-s − 24·67-s − 8·71-s − 16·74-s + 28·79-s + 5·81-s − 40·82-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s + 9-s − 1.15·12-s − 16-s − 1.41·18-s − 0.769·27-s + 3.59·31-s + 1.41·32-s + 36-s + 1.31·37-s + 3.12·41-s + 1.21·43-s + 1.15·48-s + 10/7·49-s + 2.74·53-s + 1.08·54-s − 5.08·62-s − 64-s − 2.93·67-s − 0.949·71-s − 1.85·74-s + 3.15·79-s + 5/9·81-s − 4.41·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6693168293\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6693168293\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.57884674361779277192378275215, −10.57861898893650757886255015839, −10.00926917532685268603592528672, −9.704584113283425587253663132179, −9.055172867428581192267337003432, −8.952777053694957982353977851913, −8.051205548668110708012665065461, −8.012297790898368632558712994195, −7.21049533137671014065508425666, −7.16415598959468653618847683069, −6.36746815172280571191372075579, −5.90531277092300255993086856304, −5.76845275838164810230677402759, −4.69355085039934156189379363896, −4.44193463294843333647798209742, −4.02877283626781030088204718150, −2.63746423885872204149972438473, −2.46748202581591711568639222369, −1.04615743955109003977003270597, −0.830475810933722381581175432488,
0.830475810933722381581175432488, 1.04615743955109003977003270597, 2.46748202581591711568639222369, 2.63746423885872204149972438473, 4.02877283626781030088204718150, 4.44193463294843333647798209742, 4.69355085039934156189379363896, 5.76845275838164810230677402759, 5.90531277092300255993086856304, 6.36746815172280571191372075579, 7.16415598959468653618847683069, 7.21049533137671014065508425666, 8.012297790898368632558712994195, 8.051205548668110708012665065461, 8.952777053694957982353977851913, 9.055172867428581192267337003432, 9.704584113283425587253663132179, 10.00926917532685268603592528672, 10.57861898893650757886255015839, 10.57884674361779277192378275215