| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 8-s + 9-s − 3·10-s + 12-s + 3·15-s + 16-s − 18-s + 3·20-s + 12·23-s − 24-s + 8·25-s + 27-s + 29-s − 3·30-s − 32-s + 36-s − 3·40-s − 2·43-s + 3·45-s − 12·46-s + 4·47-s + 48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s + 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.670·20-s + 2.50·23-s − 0.204·24-s + 8/5·25-s + 0.192·27-s + 0.185·29-s − 0.547·30-s − 0.176·32-s + 1/6·36-s − 0.474·40-s − 0.304·43-s + 0.447·45-s − 1.76·46-s + 0.583·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.551866599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.551866599\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.645149289840093895557147636875, −8.333414602739296745190214251611, −7.43498894585028696534192962576, −7.23488902563676555122338034850, −6.83960325421288501397922848774, −6.35238545092836530653595253820, −5.63729895810995383404509254199, −5.46878430092274319412594285838, −4.77274156204351015931789915616, −4.21038521073443578953224456230, −3.36871781260798996464363834103, −2.70855086518484661185766841901, −2.52529983238540662663608980276, −1.54566479308762826339537421413, −1.01567012886913365107839469970,
1.01567012886913365107839469970, 1.54566479308762826339537421413, 2.52529983238540662663608980276, 2.70855086518484661185766841901, 3.36871781260798996464363834103, 4.21038521073443578953224456230, 4.77274156204351015931789915616, 5.46878430092274319412594285838, 5.63729895810995383404509254199, 6.35238545092836530653595253820, 6.83960325421288501397922848774, 7.23488902563676555122338034850, 7.43498894585028696534192962576, 8.333414602739296745190214251611, 8.645149289840093895557147636875
