| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s + 2·7-s − 4·8-s + 2·9-s − 4·10-s + 8·11-s + 2·13-s − 4·14-s + 5·16-s − 4·17-s − 4·18-s − 8·19-s + 6·20-s − 16·22-s − 8·23-s + 3·25-s − 4·26-s + 6·28-s + 12·29-s − 8·31-s − 6·32-s + 8·34-s + 4·35-s + 6·36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.755·7-s − 1.41·8-s + 2/3·9-s − 1.26·10-s + 2.41·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s − 0.970·17-s − 0.942·18-s − 1.83·19-s + 1.34·20-s − 3.41·22-s − 1.66·23-s + 3/5·25-s − 0.784·26-s + 1.13·28-s + 2.22·29-s − 1.43·31-s − 1.06·32-s + 1.37·34-s + 0.676·35-s + 36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.720934177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.720934177\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 28 T + 366 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−10.24885852599419965828765454398, −9.936437107397666257424561361933, −9.342963754306563355561033919771, −9.015767737215353281332080700284, −8.612483578706019905848536563376, −8.562881720457846304559727644113, −7.949616404788202354610509543520, −7.32474856745441613976124229920, −6.87284556216085665614591093128, −6.50321026077206033475676563598, −6.11964792254415507188288844216, −6.04368470352404084154008245315, −4.98520070029572184147622599575, −4.43284176984722014183965988282, −4.00214485723677499996725200915, −3.53780078749827539939116146428, −2.32666306400481331222719247501, −2.03573334611437979415308305051, −1.54133241450534977530213345308, −0.846009753884798037301383902940,
0.846009753884798037301383902940, 1.54133241450534977530213345308, 2.03573334611437979415308305051, 2.32666306400481331222719247501, 3.53780078749827539939116146428, 4.00214485723677499996725200915, 4.43284176984722014183965988282, 4.98520070029572184147622599575, 6.04368470352404084154008245315, 6.11964792254415507188288844216, 6.50321026077206033475676563598, 6.87284556216085665614591093128, 7.32474856745441613976124229920, 7.949616404788202354610509543520, 8.562881720457846304559727644113, 8.612483578706019905848536563376, 9.015767737215353281332080700284, 9.342963754306563355561033919771, 9.936437107397666257424561361933, 10.24885852599419965828765454398
