| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s + 4·9-s − 10-s + 4·13-s − 16-s + 4·18-s + 20-s + 25-s + 4·26-s − 2·29-s + 5·32-s − 4·36-s + 8·37-s + 3·40-s + 6·41-s − 4·45-s − 8·49-s + 50-s − 4·52-s + 2·53-s − 2·58-s + 7·64-s − 4·65-s − 12·72-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s + 4/3·9-s − 0.316·10-s + 1.10·13-s − 1/4·16-s + 0.942·18-s + 0.223·20-s + 1/5·25-s + 0.784·26-s − 0.371·29-s + 0.883·32-s − 2/3·36-s + 1.31·37-s + 0.474·40-s + 0.937·41-s − 0.596·45-s − 8/7·49-s + 0.141·50-s − 0.554·52-s + 0.274·53-s − 0.262·58-s + 7/8·64-s − 0.496·65-s − 1.41·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 578000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 578000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.354799268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.354799268\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.424043056861788788961494821197, −7.946049674596831865571597876105, −7.54841064146580130342095731405, −7.07994345148301250183978142141, −6.42085190811480104298234040969, −6.19302152531675459797361142349, −5.62609126748486005743729472885, −5.05903994966954782375761579898, −4.47579008721709265204829625692, −4.24314176505705422983824620320, −3.70156366369627751113161337173, −3.27678918189018206342394694620, −2.51401669856695691459383293471, −1.57140055456994220509686407528, −0.76276492061256062585794041151,
0.76276492061256062585794041151, 1.57140055456994220509686407528, 2.51401669856695691459383293471, 3.27678918189018206342394694620, 3.70156366369627751113161337173, 4.24314176505705422983824620320, 4.47579008721709265204829625692, 5.05903994966954782375761579898, 5.62609126748486005743729472885, 6.19302152531675459797361142349, 6.42085190811480104298234040969, 7.07994345148301250183978142141, 7.54841064146580130342095731405, 7.946049674596831865571597876105, 8.424043056861788788961494821197
