| L(s) = 1 | + 6·2-s − 70·3-s − 92·4-s − 125·5-s − 420·6-s + 624·7-s − 1.32e3·8-s + 2.71e3·9-s − 750·10-s + 6.44e3·12-s + 6.82e3·13-s + 3.74e3·14-s + 8.75e3·15-s + 3.85e3·16-s + 3.62e4·17-s + 1.62e4·18-s − 3.40e4·19-s + 1.15e4·20-s − 4.36e4·21-s + 1.62e4·23-s + 9.24e4·24-s + 1.56e4·25-s + 4.09e4·26-s − 3.68e4·27-s − 5.74e4·28-s − 1.58e5·29-s + 5.25e4·30-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 1.49·3-s − 0.718·4-s − 0.447·5-s − 0.793·6-s + 0.687·7-s − 0.911·8-s + 1.24·9-s − 0.237·10-s + 1.07·12-s + 0.861·13-s + 0.364·14-s + 0.669·15-s + 0.235·16-s + 1.78·17-s + 0.657·18-s − 1.13·19-s + 0.321·20-s − 1.02·21-s + 0.279·23-s + 1.36·24-s + 1/5·25-s + 0.456·26-s − 0.360·27-s − 0.494·28-s − 1.20·29-s + 0.355·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.104907737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.104907737\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + p^{3} T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3 p T + p^{7} T^{2} \) |
| 3 | \( 1 + 70 T + p^{7} T^{2} \) |
| 7 | \( 1 - 624 T + p^{7} T^{2} \) |
| 13 | \( 1 - 6822 T + p^{7} T^{2} \) |
| 17 | \( 1 - 36246 T + p^{7} T^{2} \) |
| 19 | \( 1 + 34080 T + p^{7} T^{2} \) |
| 23 | \( 1 - 16290 T + p^{7} T^{2} \) |
| 29 | \( 1 + 158760 T + p^{7} T^{2} \) |
| 31 | \( 1 - 194620 T + p^{7} T^{2} \) |
| 37 | \( 1 - 371590 T + p^{7} T^{2} \) |
| 41 | \( 1 + 562980 T + p^{7} T^{2} \) |
| 43 | \( 1 - 234852 T + p^{7} T^{2} \) |
| 47 | \( 1 + 832530 T + p^{7} T^{2} \) |
| 53 | \( 1 + 227010 T + p^{7} T^{2} \) |
| 59 | \( 1 + 462624 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1082760 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2587910 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3097992 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2722422 T + p^{7} T^{2} \) |
| 79 | \( 1 - 211620 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5216628 T + p^{7} T^{2} \) |
| 89 | \( 1 + 9077130 T + p^{7} T^{2} \) |
| 97 | \( 1 - 14734790 T + p^{7} T^{2} \) |
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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
−9.739548064978092068007439556101, −8.523376311889470201553187325217, −7.79370610137532418181483641115, −6.44836063850120351244728979708, −5.76367171275450426391453451558, −4.99602127145746358106007876366, −4.28081684196616174745012448829, −3.29031902905103857430263759940, −1.37099100568690697799523626355, −0.49943586700642653917223753955,
0.49943586700642653917223753955, 1.37099100568690697799523626355, 3.29031902905103857430263759940, 4.28081684196616174745012448829, 4.99602127145746358106007876366, 5.76367171275450426391453451558, 6.44836063850120351244728979708, 7.79370610137532418181483641115, 8.523376311889470201553187325217, 9.739548064978092068007439556101
