| L(s) = 1 | − 3-s − 3·5-s − 2·9-s + 2·11-s + 3·15-s + 4·23-s + 4·25-s + 5·27-s + 5·31-s − 2·33-s + 6·37-s + 6·45-s + 4·47-s − 10·49-s + 5·53-s − 6·55-s − 6·59-s − 23·67-s − 4·69-s − 7·71-s − 4·75-s + 81-s − 13·89-s − 5·93-s − 4·97-s − 4·99-s + 19·103-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 2/3·9-s + 0.603·11-s + 0.774·15-s + 0.834·23-s + 4/5·25-s + 0.962·27-s + 0.898·31-s − 0.348·33-s + 0.986·37-s + 0.894·45-s + 0.583·47-s − 1.42·49-s + 0.686·53-s − 0.809·55-s − 0.781·59-s − 2.80·67-s − 0.481·69-s − 0.830·71-s − 0.461·75-s + 1/9·81-s − 1.37·89-s − 0.518·93-s − 0.406·97-s − 0.402·99-s + 1.87·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 69 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 114 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.69523355267074010579353379643, −7.15288637724728766910966645651, −6.84075322671062874923353429951, −6.21380074708463504538051723697, −6.03073759342737771610754526197, −5.47490891283532515893036163992, −4.82364209015431528910892167058, −4.54753896713290821254528397175, −4.13814620487421848486060053197, −3.51557774992649243759247430532, −2.99769703021923011838172046712, −2.66549272167519540435260765387, −1.57992977830527588285747630766, −0.854292478382882462353343449317, 0,
0.854292478382882462353343449317, 1.57992977830527588285747630766, 2.66549272167519540435260765387, 2.99769703021923011838172046712, 3.51557774992649243759247430532, 4.13814620487421848486060053197, 4.54753896713290821254528397175, 4.82364209015431528910892167058, 5.47490891283532515893036163992, 6.03073759342737771610754526197, 6.21380074708463504538051723697, 6.84075322671062874923353429951, 7.15288637724728766910966645651, 7.69523355267074010579353379643
