| L(s) = 1 | − 16·2-s + 81·3-s + 256·4-s − 625·5-s − 1.29e3·6-s + 7.19e3·7-s − 4.09e3·8-s + 6.56e3·9-s + 1.00e4·10-s − 4.56e4·11-s + 2.07e4·12-s − 3.40e4·13-s − 1.15e5·14-s − 5.06e4·15-s + 6.55e4·16-s + 4.24e5·17-s − 1.04e5·18-s + 9.30e5·19-s − 1.60e5·20-s + 5.82e5·21-s + 7.29e5·22-s + 1.57e6·23-s − 3.31e5·24-s + 3.90e5·25-s + 5.45e5·26-s + 5.31e5·27-s + 1.84e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.939·11-s + 0.288·12-s − 0.330·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.23·17-s − 0.235·18-s + 1.63·19-s − 0.223·20-s + 0.654·21-s + 0.664·22-s + 1.17·23-s − 0.204·24-s + 1/5·25-s + 0.233·26-s + 0.192·27-s + 0.566·28-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
| \(L(5)\) |
\(\approx\) |
\(1.704964451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704964451\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 1028 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 45600 T + p^{9} T^{2} \) |
| 13 | \( 1 + 34078 T + p^{9} T^{2} \) |
| 17 | \( 1 - 424326 T + p^{9} T^{2} \) |
| 19 | \( 1 - 930716 T + p^{9} T^{2} \) |
| 23 | \( 1 - 1572000 T + p^{9} T^{2} \) |
| 29 | \( 1 + 864486 T + p^{9} T^{2} \) |
| 31 | \( 1 - 8703368 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7526878 T + p^{9} T^{2} \) |
| 41 | \( 1 - 8562234 T + p^{9} T^{2} \) |
| 43 | \( 1 - 32831996 T + p^{9} T^{2} \) |
| 47 | \( 1 + 38536800 T + p^{9} T^{2} \) |
| 53 | \( 1 + 35746086 T + p^{9} T^{2} \) |
| 59 | \( 1 + 77109600 T + p^{9} T^{2} \) |
| 61 | \( 1 - 18400790 T + p^{9} T^{2} \) |
| 67 | \( 1 + 142510084 T + p^{9} T^{2} \) |
| 71 | \( 1 - 318643200 T + p^{9} T^{2} \) |
| 73 | \( 1 + 30899518 T + p^{9} T^{2} \) |
| 79 | \( 1 - 603013448 T + p^{9} T^{2} \) |
| 83 | \( 1 + 493844148 T + p^{9} T^{2} \) |
| 89 | \( 1 + 92882862 T + p^{9} T^{2} \) |
| 97 | \( 1 + 755725438 T + p^{9} T^{2} \) |
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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
−15.06326510601418622809159489445, −13.95051175285088843774526668124, −12.23776974550312668595683686262, −11.01206987575214618807634235881, −9.666603404397369085860784432838, −8.152671849562915419173326166771, −7.47706475094358706311570516170, −5.08606514072376439525867334282, −2.94469963642540191993849225605, −1.12265695968181164556022976572,
1.12265695968181164556022976572, 2.94469963642540191993849225605, 5.08606514072376439525867334282, 7.47706475094358706311570516170, 8.152671849562915419173326166771, 9.666603404397369085860784432838, 11.01206987575214618807634235881, 12.23776974550312668595683686262, 13.95051175285088843774526668124, 15.06326510601418622809159489445
