L(s) = 1 | + 2-s + 4-s + 5-s − 8-s − 4·9-s + 10-s + 5·11-s − 3·16-s + 9·17-s − 4·18-s + 20-s + 5·22-s − 3·25-s − 5·32-s + 9·34-s − 4·36-s − 40-s + 5·44-s − 4·45-s − 14·47-s − 49-s − 3·50-s − 17·53-s + 5·55-s − 3·64-s + 6·67-s + 9·68-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 4/3·9-s + 0.316·10-s + 1.50·11-s − 3/4·16-s + 2.18·17-s − 0.942·18-s + 0.223·20-s + 1.06·22-s − 3/5·25-s − 0.883·32-s + 1.54·34-s − 2/3·36-s − 0.158·40-s + 0.753·44-s − 0.596·45-s − 2.04·47-s − 1/7·49-s − 0.424·50-s − 2.33·53-s + 0.674·55-s − 3/8·64-s + 0.733·67-s + 1.09·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15842 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15842 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.664295681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.664295681\) |
\(L(\frac{3}{2})\) |
|
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_1$$\times$$C_2$ | \( ( 1 + T )( 1 - p T + p T^{2} ) \) |
| 89 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−11.36256181879199372961864697868, −10.68385138209229190917850337844, −9.809164145641349077396594833304, −9.572558499423294552564299053641, −8.945836535217341721093195996343, −8.296741473353169824858822633472, −7.75456583606896705833766076180, −6.87867502477427305358091343102, −6.24616920554874021767754263929, −5.86249394904599278160568029053, −5.33491583183369725841938108131, −4.43830843924901719623986028675, −3.34228072249885749569002547896, −3.13163524465221465375217618833, −1.70581462658404761171497866648,
1.70581462658404761171497866648, 3.13163524465221465375217618833, 3.34228072249885749569002547896, 4.43830843924901719623986028675, 5.33491583183369725841938108131, 5.86249394904599278160568029053, 6.24616920554874021767754263929, 6.87867502477427305358091343102, 7.75456583606896705833766076180, 8.296741473353169824858822633472, 8.945836535217341721093195996343, 9.572558499423294552564299053641, 9.809164145641349077396594833304, 10.68385138209229190917850337844, 11.36256181879199372961864697868