| L(s) = 1 | + 4·2-s + 16·4-s + 91·5-s − 49·7-s + 64·8-s + 364·10-s − 61·11-s + 156·13-s − 196·14-s + 256·16-s + 614·17-s + 2.20e3·19-s + 1.45e3·20-s − 244·22-s − 3.13e3·23-s + 5.15e3·25-s + 624·26-s − 784·28-s + 3.42e3·29-s + 6.43e3·31-s + 1.02e3·32-s + 2.45e3·34-s − 4.45e3·35-s − 6.19e3·37-s + 8.82e3·38-s + 5.82e3·40-s + 4.92e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.62·5-s − 0.377·7-s + 0.353·8-s + 1.15·10-s − 0.152·11-s + 0.256·13-s − 0.267·14-s + 1/4·16-s + 0.515·17-s + 1.40·19-s + 0.813·20-s − 0.107·22-s − 1.23·23-s + 1.64·25-s + 0.181·26-s − 0.188·28-s + 0.756·29-s + 1.20·31-s + 0.176·32-s + 0.364·34-s − 0.615·35-s − 0.744·37-s + 0.991·38-s + 0.575·40-s + 0.457·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.820639816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.820639816\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 91 T + p^{5} T^{2} \) |
| 11 | \( 1 + 61 T + p^{5} T^{2} \) |
| 13 | \( 1 - 12 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 614 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2207 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3139 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3424 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6435 T + p^{5} T^{2} \) |
| 37 | \( 1 + 6199 T + p^{5} T^{2} \) |
| 41 | \( 1 - 4929 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4222 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5142 T + p^{5} T^{2} \) |
| 53 | \( 1 - 108 p T + p^{5} T^{2} \) |
| 59 | \( 1 + 15902 T + p^{5} T^{2} \) |
| 61 | \( 1 - 18624 T + p^{5} T^{2} \) |
| 67 | \( 1 - 11884 T + p^{5} T^{2} \) |
| 71 | \( 1 + 55879 T + p^{5} T^{2} \) |
| 73 | \( 1 + 42494 T + p^{5} T^{2} \) |
| 79 | \( 1 + 23622 T + p^{5} T^{2} \) |
| 83 | \( 1 - 79400 T + p^{5} T^{2} \) |
| 89 | \( 1 - 12201 T + p^{5} T^{2} \) |
| 97 | \( 1 - 104080 T + p^{5} T^{2} \) |
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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
−10.22271088386812030932476048164, −9.988429748914251152040098724532, −8.839640754675954626287929495562, −7.52775913054123855198008728023, −6.34535451328317193309770600256, −5.78520608232576594612004917440, −4.84574775442511437492265519200, −3.34921616730732833921538755091, −2.32440657024443934364003324919, −1.15053780621232406488103693215,
1.15053780621232406488103693215, 2.32440657024443934364003324919, 3.34921616730732833921538755091, 4.84574775442511437492265519200, 5.78520608232576594612004917440, 6.34535451328317193309770600256, 7.52775913054123855198008728023, 8.839640754675954626287929495562, 9.988429748914251152040098724532, 10.22271088386812030932476048164
