| L(s) = 1 | − 16·2-s − 81·3-s + 256·4-s + 625·5-s + 1.29e3·6-s − 7.16e3·7-s − 4.09e3·8-s + 6.56e3·9-s − 1.00e4·10-s − 8.37e4·11-s − 2.07e4·12-s + 1.28e5·13-s + 1.14e5·14-s − 5.06e4·15-s + 6.55e4·16-s + 5.60e5·17-s − 1.04e5·18-s − 5.77e5·19-s + 1.60e5·20-s + 5.80e5·21-s + 1.33e6·22-s + 2.43e6·23-s + 3.31e5·24-s + 3.90e5·25-s − 2.05e6·26-s − 5.31e5·27-s − 1.83e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.12·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.72·11-s − 0.288·12-s + 1.24·13-s + 0.797·14-s − 0.258·15-s + 1/4·16-s + 1.62·17-s − 0.235·18-s − 1.01·19-s + 0.223·20-s + 0.651·21-s + 1.21·22-s + 1.81·23-s + 0.204·24-s + 1/5·25-s − 0.879·26-s − 0.192·27-s − 0.564·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9295612915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9295612915\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 3 | \( 1 + p^{4} T \) |
| 5 | \( 1 - p^{4} T \) |
| good | 7 | \( 1 + 1024 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 83748 T + p^{9} T^{2} \) |
| 13 | \( 1 - 128126 T + p^{9} T^{2} \) |
| 17 | \( 1 - 560802 T + p^{9} T^{2} \) |
| 19 | \( 1 + 577660 T + p^{9} T^{2} \) |
| 23 | \( 1 - 2431296 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5791710 T + p^{9} T^{2} \) |
| 31 | \( 1 - 4145312 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7011658 T + p^{9} T^{2} \) |
| 41 | \( 1 + 8881398 T + p^{9} T^{2} \) |
| 43 | \( 1 + 15730684 T + p^{9} T^{2} \) |
| 47 | \( 1 - 60552072 T + p^{9} T^{2} \) |
| 53 | \( 1 - 30273366 T + p^{9} T^{2} \) |
| 59 | \( 1 - 45957660 T + p^{9} T^{2} \) |
| 61 | \( 1 - 37595102 T + p^{9} T^{2} \) |
| 67 | \( 1 - 196784012 T + p^{9} T^{2} \) |
| 71 | \( 1 - 56047992 T + p^{9} T^{2} \) |
| 73 | \( 1 + 159688054 T + p^{9} T^{2} \) |
| 79 | \( 1 - 201923360 T + p^{9} T^{2} \) |
| 83 | \( 1 + 362955444 T + p^{9} T^{2} \) |
| 89 | \( 1 + 272479110 T + p^{9} T^{2} \) |
| 97 | \( 1 + 600852478 T + p^{9} T^{2} \) |
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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
−15.44398457559922005190567987167, −13.42810623082706486503143129401, −12.46918708599500910838666811229, −10.71949828223043690229536707576, −10.02970797175950460500927810410, −8.430206185924368549545379739688, −6.76052359436354631217103418528, −5.51489763408868537117784422382, −2.95791996846829319612594429973, −0.803764311654455407745487431651,
0.803764311654455407745487431651, 2.95791996846829319612594429973, 5.51489763408868537117784422382, 6.76052359436354631217103418528, 8.430206185924368549545379739688, 10.02970797175950460500927810410, 10.71949828223043690229536707576, 12.46918708599500910838666811229, 13.42810623082706486503143129401, 15.44398457559922005190567987167
