| L(s) = 1 | + 5-s + 9-s + 3·13-s − 4·17-s − 4·25-s + 29-s + 37-s − 10·41-s + 45-s − 5·49-s − 14·53-s + 11·61-s + 3·65-s − 16·73-s + 81-s − 4·85-s − 5·89-s + 8·97-s − 11·101-s + 16·109-s − 18·113-s + 3·117-s − 20·121-s − 9·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1/3·9-s + 0.832·13-s − 0.970·17-s − 4/5·25-s + 0.185·29-s + 0.164·37-s − 1.56·41-s + 0.149·45-s − 5/7·49-s − 1.92·53-s + 1.40·61-s + 0.372·65-s − 1.87·73-s + 1/9·81-s − 0.433·85-s − 0.529·89-s + 0.812·97-s − 1.09·101-s + 1.53·109-s − 1.69·113-s + 0.277·117-s − 1.81·121-s − 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.007572479786186429959160263097, −7.50775960911915889165908657292, −7.01134759773254982162177405940, −6.49122435891094706409615970761, −6.30305085881165228803995293587, −5.77240404576727640453247320266, −5.18961258432143749421947160041, −4.79549074202612395897022020829, −4.20178055748563691427329453500, −3.75687132410857773455862627584, −3.17241906978367924630604803558, −2.52272590326081379539411314065, −1.81086296735501049212536358357, −1.33074763446168462593334620467, 0,
1.33074763446168462593334620467, 1.81086296735501049212536358357, 2.52272590326081379539411314065, 3.17241906978367924630604803558, 3.75687132410857773455862627584, 4.20178055748563691427329453500, 4.79549074202612395897022020829, 5.18961258432143749421947160041, 5.77240404576727640453247320266, 6.30305085881165228803995293587, 6.49122435891094706409615970761, 7.01134759773254982162177405940, 7.50775960911915889165908657292, 8.007572479786186429959160263097
