| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 3·11-s + 12-s + 16-s + 9·17-s − 18-s + 19-s − 3·22-s − 24-s + 2·25-s + 27-s − 32-s + 3·33-s − 9·34-s + 36-s − 38-s + 6·41-s − 11·43-s + 3·44-s + 48-s − 13·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s + 2.18·17-s − 0.235·18-s + 0.229·19-s − 0.639·22-s − 0.204·24-s + 2/5·25-s + 0.192·27-s − 0.176·32-s + 0.522·33-s − 1.54·34-s + 1/6·36-s − 0.162·38-s + 0.937·41-s − 1.67·43-s + 0.452·44-s + 0.144·48-s − 1.85·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.537778986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.537778986\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.747707191175209441218296822071, −9.113408463243647775134903010279, −8.695348607584860387911326116543, −8.123028121475115051227077007196, −7.70776955757295169857288579509, −7.34800435577449555843449973696, −6.61839689115813819100372203397, −6.23252991734223231797622937812, −5.52545831247434625509168219034, −4.92310656236982106258656946630, −4.12786989261980462344842823209, −3.27652432740011333824788859201, −3.09975095477689994772071629147, −1.85278941636313118481788942622, −1.13139588167605795155122494146,
1.13139588167605795155122494146, 1.85278941636313118481788942622, 3.09975095477689994772071629147, 3.27652432740011333824788859201, 4.12786989261980462344842823209, 4.92310656236982106258656946630, 5.52545831247434625509168219034, 6.23252991734223231797622937812, 6.61839689115813819100372203397, 7.34800435577449555843449973696, 7.70776955757295169857288579509, 8.123028121475115051227077007196, 8.695348607584860387911326116543, 9.113408463243647775134903010279, 9.747707191175209441218296822071
