L(s) = 1 | − 4-s + 7-s + 16-s − 6·25-s − 28-s + 20·37-s + 8·43-s + 49-s − 64-s + 8·67-s + 6·100-s + 4·109-s + 112-s + 6·121-s + 127-s + 131-s + 137-s + 139-s − 20·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 8·172-s + 173-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 0.377·7-s + 1/4·16-s − 6/5·25-s − 0.188·28-s + 3.28·37-s + 1.21·43-s + 1/7·49-s − 1/8·64-s + 0.977·67-s + 3/5·100-s + 0.383·109-s + 0.0944·112-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.64·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s − 0.609·172-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.381015326\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381015326\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.453505489276972732329133349760, −9.164357727767744996554627058200, −8.453234310603238542956108613739, −7.910181575788804740460685750914, −7.77698474257649382795973883033, −7.07715851976662932792536475690, −6.39550603345587366482416300943, −5.82638850103906378738309356342, −5.51933717252283675583847570281, −4.59219535120872901055676782946, −4.33214489367502481515592583866, −3.67417741905446743355217837016, −2.79160913843474065678897118940, −2.07080373981098315618079743058, −0.883484333734674189049376948209,
0.883484333734674189049376948209, 2.07080373981098315618079743058, 2.79160913843474065678897118940, 3.67417741905446743355217837016, 4.33214489367502481515592583866, 4.59219535120872901055676782946, 5.51933717252283675583847570281, 5.82638850103906378738309356342, 6.39550603345587366482416300943, 7.07715851976662932792536475690, 7.77698474257649382795973883033, 7.910181575788804740460685750914, 8.453234310603238542956108613739, 9.164357727767744996554627058200, 9.453505489276972732329133349760