| L(s) = 1 | − 2-s + 4-s − 8-s + 16-s + 6·19-s + 4·25-s − 8·29-s − 32-s − 6·38-s − 4·41-s − 10·49-s − 4·50-s + 4·53-s + 8·58-s + 12·59-s + 12·61-s + 64-s − 20·71-s + 4·73-s + 6·76-s + 4·82-s + 8·89-s + 10·98-s + 4·100-s − 4·106-s + 20·107-s + 36·113-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/4·16-s + 1.37·19-s + 4/5·25-s − 1.48·29-s − 0.176·32-s − 0.973·38-s − 0.624·41-s − 1.42·49-s − 0.565·50-s + 0.549·53-s + 1.05·58-s + 1.56·59-s + 1.53·61-s + 1/8·64-s − 2.37·71-s + 0.468·73-s + 0.688·76-s + 0.441·82-s + 0.847·89-s + 1.01·98-s + 2/5·100-s − 0.388·106-s + 1.93·107-s + 3.38·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265174550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265174550\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 36 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
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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
−8.516879814018461630582698708477, −8.266657033166355627224634461617, −7.54404579865140564036457014922, −7.20737828319810407767603293925, −7.03246628017271905239827966315, −6.20603950122645105978854664675, −5.90938504817634015506463323428, −5.19936610432714340308691823288, −4.97399563196003547290327956155, −4.12585325324904006322529950852, −3.46940621813603015777625007687, −3.10132945352665839719719210344, −2.26813573409953358871707952545, −1.60699522957701214283534483367, −0.69428874128268000137987413707,
0.69428874128268000137987413707, 1.60699522957701214283534483367, 2.26813573409953358871707952545, 3.10132945352665839719719210344, 3.46940621813603015777625007687, 4.12585325324904006322529950852, 4.97399563196003547290327956155, 5.19936610432714340308691823288, 5.90938504817634015506463323428, 6.20603950122645105978854664675, 7.03246628017271905239827966315, 7.20737828319810407767603293925, 7.54404579865140564036457014922, 8.266657033166355627224634461617, 8.516879814018461630582698708477
