| L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s − 7-s + 3·8-s + 9-s − 2·10-s + 11-s + 12-s + 6·13-s + 14-s − 2·15-s − 16-s − 2·17-s − 18-s − 4·19-s − 2·20-s + 21-s − 22-s − 3·24-s − 25-s − 6·26-s − 27-s + 28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s + 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.213·22-s − 0.612·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122199 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122199 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.82219625728816, −13.18203879958063, −12.87092658806111, −12.62450928616991, −11.54483926151592, −11.36025295527716, −10.73250753352786, −10.35286424236161, −9.856806123772935, −9.450592278925831, −8.962429789466412, −8.393675128950850, −8.191414551603139, −7.331763703031171, −6.636924482058393, −6.403786281617044, −5.788784041130661, −5.397411396710102, −4.610335249182316, −4.086552548687128, −3.707407460293909, −2.750511490649688, −1.984271032672823, −1.400809249393244, −0.8405053847280338, 0,
0.8405053847280338, 1.400809249393244, 1.984271032672823, 2.750511490649688, 3.707407460293909, 4.086552548687128, 4.610335249182316, 5.397411396710102, 5.788784041130661, 6.403786281617044, 6.636924482058393, 7.331763703031171, 8.191414551603139, 8.393675128950850, 8.962429789466412, 9.450592278925831, 9.856806123772935, 10.35286424236161, 10.73250753352786, 11.36025295527716, 11.54483926151592, 12.62450928616991, 12.87092658806111, 13.18203879958063, 13.82219625728816
