| L(s) = 1 | − 2·2-s − 3·3-s + 2·4-s + 6·6-s + 6·9-s − 10·11-s − 6·12-s − 4·13-s − 4·16-s − 12·18-s + 20·22-s + 4·23-s − 6·25-s + 8·26-s − 9·27-s + 8·32-s + 30·33-s + 12·36-s − 2·37-s + 12·39-s − 20·44-s − 8·46-s − 18·47-s + 12·48-s − 13·49-s + 12·50-s − 8·52-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 4-s + 2.44·6-s + 2·9-s − 3.01·11-s − 1.73·12-s − 1.10·13-s − 16-s − 2.82·18-s + 4.26·22-s + 0.834·23-s − 6/5·25-s + 1.56·26-s − 1.73·27-s + 1.41·32-s + 5.22·33-s + 2·36-s − 0.328·37-s + 1.92·39-s − 3.01·44-s − 1.17·46-s − 2.62·47-s + 1.73·48-s − 1.85·49-s + 1.69·50-s − 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−8.411523075727323097878765523060, −8.014330807872879223418899196928, −7.85187058889734411022222226313, −7.20420824172938396254941904877, −6.87039121695443194852465216240, −6.30588286042335336353260552535, −5.52489447686531677812445208304, −5.10147923434552743490630958114, −5.00317001400665869534627315571, −4.25615660952913416427789271504, −3.02975166758349613305737908394, −2.38413465275342115890308840083, −1.51328298133418770573309670037, 0, 0,
1.51328298133418770573309670037, 2.38413465275342115890308840083, 3.02975166758349613305737908394, 4.25615660952913416427789271504, 5.00317001400665869534627315571, 5.10147923434552743490630958114, 5.52489447686531677812445208304, 6.30588286042335336353260552535, 6.87039121695443194852465216240, 7.20420824172938396254941904877, 7.85187058889734411022222226313, 8.014330807872879223418899196928, 8.411523075727323097878765523060
