| L(s) = 1 | + 2-s − 3-s − 6-s − 2·7-s − 8-s + 3·9-s − 12·11-s − 5·13-s − 2·14-s − 16-s − 3·17-s + 3·18-s + 2·21-s − 12·22-s − 3·23-s + 24-s + 5·25-s − 5·26-s − 8·27-s − 9·29-s − 8·31-s + 12·33-s − 3·34-s + 4·37-s + 5·39-s + 2·42-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 9-s − 3.61·11-s − 1.38·13-s − 0.534·14-s − 1/4·16-s − 0.727·17-s + 0.707·18-s + 0.436·21-s − 2.55·22-s − 0.625·23-s + 0.204·24-s + 25-s − 0.980·26-s − 1.53·27-s − 1.67·29-s − 1.43·31-s + 2.08·33-s − 0.514·34-s + 0.657·37-s + 0.800·39-s + 0.308·42-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−10.25765001529881292714042737125, −9.886190533135730460502822307684, −9.446100495936444896176596385838, −9.169466048648994520446296461214, −8.092572216291284359554443608429, −8.056905948556871485554174373473, −7.49553318287091591647380974186, −7.12814032116340885760729957749, −6.72641471866208949390230551007, −5.94820683754930139030370328507, −5.53479944941460380950923957990, −5.23918886731926048487804639182, −4.79104111464750286928878710427, −4.46515944968277356981876872342, −3.51265975921653664699823417975, −3.17373830502963259637811322791, −2.20643893612872520195976331227, −2.19454637273018313789251851896, 0, 0,
2.19454637273018313789251851896, 2.20643893612872520195976331227, 3.17373830502963259637811322791, 3.51265975921653664699823417975, 4.46515944968277356981876872342, 4.79104111464750286928878710427, 5.23918886731926048487804639182, 5.53479944941460380950923957990, 5.94820683754930139030370328507, 6.72641471866208949390230551007, 7.12814032116340885760729957749, 7.49553318287091591647380974186, 8.056905948556871485554174373473, 8.092572216291284359554443608429, 9.169466048648994520446296461214, 9.446100495936444896176596385838, 9.886190533135730460502822307684, 10.25765001529881292714042737125
