| L(s) = 1 | − 2·4-s − 3·5-s + 9-s + 6·20-s − 3·23-s + 4·25-s + 4·31-s − 2·36-s + 6·37-s − 3·45-s − 9·47-s − 4·49-s − 18·53-s + 3·59-s + 8·64-s + 24·67-s − 3·71-s + 81-s − 3·89-s + 6·92-s − 8·100-s + 6·103-s − 3·113-s + 9·115-s − 8·124-s + 3·125-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s + 1/3·9-s + 1.34·20-s − 0.625·23-s + 4/5·25-s + 0.718·31-s − 1/3·36-s + 0.986·37-s − 0.447·45-s − 1.31·47-s − 4/7·49-s − 2.47·53-s + 0.390·59-s + 64-s + 2.93·67-s − 0.356·71-s + 1/9·81-s − 0.317·89-s + 0.625·92-s − 4/5·100-s + 0.591·103-s − 0.282·113-s + 0.839·115-s − 0.718·124-s + 0.268·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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
−7.47940989837764363954032281192, −6.80263715871571754171837420106, −6.61652829171063868440110633063, −6.12057939698499850735830881946, −5.55845115780048886607875656061, −4.90018075076443130021167442427, −4.74526111593017959386003359162, −4.36344754882561200494685590013, −3.78539727127252690371054728720, −3.59817831642777593627469229477, −2.96785743114780570698224297243, −2.31969335712321486534787457423, −1.54678956680124712333616942355, −0.70948635539042139653477101799, 0,
0.70948635539042139653477101799, 1.54678956680124712333616942355, 2.31969335712321486534787457423, 2.96785743114780570698224297243, 3.59817831642777593627469229477, 3.78539727127252690371054728720, 4.36344754882561200494685590013, 4.74526111593017959386003359162, 4.90018075076443130021167442427, 5.55845115780048886607875656061, 6.12057939698499850735830881946, 6.61652829171063868440110633063, 6.80263715871571754171837420106, 7.47940989837764363954032281192
