L(s) = 1 | − 1.51e3·5-s + 1.02e4·7-s − 3.91e3·11-s − 1.76e5·13-s − 1.48e5·17-s + 4.99e5·19-s + 1.88e6·23-s + 3.26e5·25-s + 9.20e5·29-s + 1.37e6·31-s − 1.54e7·35-s + 5.06e6·37-s + 2.41e7·41-s + 2.57e7·43-s + 6.07e7·47-s + 6.46e7·49-s − 2.94e7·53-s + 5.91e6·55-s − 5.18e7·59-s + 3.34e7·61-s + 2.66e8·65-s + 1.44e8·67-s − 6.83e7·71-s + 1.68e8·73-s − 4.01e7·77-s + 2.35e8·79-s + 6.46e7·83-s + ⋯ |
L(s) = 1 | − 1.08·5-s + 1.61·7-s − 0.0806·11-s − 1.71·13-s − 0.430·17-s + 0.879·19-s + 1.40·23-s + 0.167·25-s + 0.241·29-s + 0.268·31-s − 1.74·35-s + 0.444·37-s + 1.33·41-s + 1.15·43-s + 1.81·47-s + 1.60·49-s − 0.513·53-s + 0.0871·55-s − 0.556·59-s + 0.309·61-s + 1.85·65-s + 0.878·67-s − 0.319·71-s + 0.693·73-s − 0.130·77-s + 0.679·79-s + 0.149·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.792932606\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.792932606\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 302 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 1464 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 356 p T + p^{9} T^{2} \) |
| 13 | \( 1 + 176594 T + p^{9} T^{2} \) |
| 17 | \( 1 + 148370 T + p^{9} T^{2} \) |
| 19 | \( 1 - 499796 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1889768 T + p^{9} T^{2} \) |
| 29 | \( 1 - 920898 T + p^{9} T^{2} \) |
| 31 | \( 1 - 1379360 T + p^{9} T^{2} \) |
| 37 | \( 1 - 5064966 T + p^{9} T^{2} \) |
| 41 | \( 1 - 24100758 T + p^{9} T^{2} \) |
| 43 | \( 1 - 25785196 T + p^{9} T^{2} \) |
| 47 | \( 1 - 60790224 T + p^{9} T^{2} \) |
| 53 | \( 1 + 29496214 T + p^{9} T^{2} \) |
| 59 | \( 1 + 51819388 T + p^{9} T^{2} \) |
| 61 | \( 1 - 33426910 T + p^{9} T^{2} \) |
| 67 | \( 1 - 144856196 T + p^{9} T^{2} \) |
| 71 | \( 1 + 68397128 T + p^{9} T^{2} \) |
| 73 | \( 1 - 168216202 T + p^{9} T^{2} \) |
| 79 | \( 1 - 235398736 T + p^{9} T^{2} \) |
| 83 | \( 1 - 64639852 T + p^{9} T^{2} \) |
| 89 | \( 1 - 78782694 T + p^{9} T^{2} \) |
| 97 | \( 1 + 24113566 T + p^{9} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.46933442584242213011816443078, −11.62217987382810386475230533845, −10.82391912791471530814374786300, −9.221585923463243730985253330641, −7.87628317785162522878546930892, −7.32159061804715983991206343461, −5.17084873637035459281473908671, −4.31631041957734761300249783568, −2.51639796757233192113176605660, −0.816868918067179954054785609355,
0.816868918067179954054785609355, 2.51639796757233192113176605660, 4.31631041957734761300249783568, 5.17084873637035459281473908671, 7.32159061804715983991206343461, 7.87628317785162522878546930892, 9.221585923463243730985253330641, 10.82391912791471530814374786300, 11.62217987382810386475230533845, 12.46933442584242213011816443078