L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 7-s + 8-s − 10-s + 11-s − 2·13-s + 14-s − 15-s − 16-s + 6·17-s + 4·19-s + 21-s − 22-s + 6·23-s − 24-s + 5·25-s + 2·26-s + 27-s + 6·29-s + 30-s + 11·31-s − 33-s − 6·34-s − 35-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.218·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 25-s + 0.392·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 1.97·31-s − 0.174·33-s − 1.02·34-s − 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.154056637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.154056637\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5 T - 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−10.79292762452724638664465111222, −10.59968536796956491800518294361, −9.941274010020743059097259597581, −9.820929896446834318321933931817, −9.399962436611779990352195139564, −8.867364660409077181261747633233, −8.440177060108751622383883953505, −7.969137781154364086106483468282, −7.22440246136445821184965402644, −7.22346425429358204649414479361, −6.42425224506268994397092301196, −6.05502022869840190829460552562, −5.53074109270997193839021520966, −4.84520803212155032273063532775, −4.71654687659452427119489782948, −3.76773780067947046869286477055, −2.83931838297286137220896364174, −2.78934340961851449924498546850, −1.29709934570008451130050918403, −0.871446442840109719682469124242,
0.871446442840109719682469124242, 1.29709934570008451130050918403, 2.78934340961851449924498546850, 2.83931838297286137220896364174, 3.76773780067947046869286477055, 4.71654687659452427119489782948, 4.84520803212155032273063532775, 5.53074109270997193839021520966, 6.05502022869840190829460552562, 6.42425224506268994397092301196, 7.22346425429358204649414479361, 7.22440246136445821184965402644, 7.969137781154364086106483468282, 8.440177060108751622383883953505, 8.867364660409077181261747633233, 9.399962436611779990352195139564, 9.820929896446834318321933931817, 9.941274010020743059097259597581, 10.59968536796956491800518294361, 10.79292762452724638664465111222