| L(s) = 1 | + 2-s + 4-s + 8-s − 5·9-s − 12·11-s − 10·13-s + 16-s + 6·17-s − 5·18-s + 19-s − 12·22-s − 10·25-s − 10·26-s − 18·29-s + 8·31-s + 32-s + 6·34-s − 5·36-s − 4·37-s + 38-s + 16·43-s − 12·44-s − 13·49-s − 10·50-s − 10·52-s + 6·53-s − 18·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 5/3·9-s − 3.61·11-s − 2.77·13-s + 1/4·16-s + 1.45·17-s − 1.17·18-s + 0.229·19-s − 2.55·22-s − 2·25-s − 1.96·26-s − 3.34·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s − 5/6·36-s − 0.657·37-s + 0.162·38-s + 2.43·43-s − 1.80·44-s − 1.85·49-s − 1.41·50-s − 1.38·52-s + 0.824·53-s − 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 877952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.87022552956968366194501622566, −7.40476303111443747598071967849, −7.14449686564876148233582580493, −6.09727019472392450239856120896, −5.56559135650201084358170066267, −5.41480429146700181457700017195, −5.32600053322851104543262796765, −4.76456060038860011441331940251, −3.98931312476793193392332292270, −3.24158791522542483628555695808, −2.80468657266648836263703727877, −2.38005475536401603079615670633, −2.16004950277651645615210004057, 0, 0,
2.16004950277651645615210004057, 2.38005475536401603079615670633, 2.80468657266648836263703727877, 3.24158791522542483628555695808, 3.98931312476793193392332292270, 4.76456060038860011441331940251, 5.32600053322851104543262796765, 5.41480429146700181457700017195, 5.56559135650201084358170066267, 6.09727019472392450239856120896, 7.14449686564876148233582580493, 7.40476303111443747598071967849, 7.87022552956968366194501622566
