| L(s) = 1 | + 7.48·5-s − 13.4·7-s + 11.9·11-s + 31.0·25-s + 50·29-s + 61.4·31-s − 100.·35-s + 132.·49-s + 4.57·53-s + 89.6·55-s + 10·59-s − 143.·73-s − 161.·77-s + 58·79-s − 17.8·83-s + 128.·97-s + 154.·101-s + 10·103-s + 212.·107-s + ⋯ |
| L(s) = 1 | + 1.49·5-s − 1.92·7-s + 1.08·11-s + 1.24·25-s + 1.72·29-s + 1.98·31-s − 2.88·35-s + 2.71·49-s + 0.0862·53-s + 1.62·55-s + 0.169·59-s − 1.96·73-s − 2.09·77-s + 0.734·79-s − 0.215·83-s + 1.32·97-s + 1.52·101-s + 0.0970·103-s + 1.98·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.165699392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.165699392\) |
| \(L(2)\) |
|
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 | \( 1 - 7.48T + 25T^{2} \) |
| 7 | \( 1 + 13.4T + 49T^{2} \) |
| 11 | \( 1 - 11.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 50T + 841T^{2} \) |
| 31 | \( 1 - 61.4T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.57T + 2.80e3T^{2} \) |
| 59 | \( 1 - 10T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 143.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58T + 6.24e3T^{2} \) |
| 83 | \( 1 + 17.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 128.T + 9.40e3T^{2} \) |
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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
−10.06075186511197498786992640977, −9.280162239023777035467651422160, −8.627913734733030451301225583250, −7.02657822281629818148146991200, −6.28165334955427960098520591649, −6.04347984886334806470054103310, −4.62251208921023152475242741351, −3.33028297615826984181789921790, −2.46398985917455291624305097961, −0.974323119477032501061001207524,
0.974323119477032501061001207524, 2.46398985917455291624305097961, 3.33028297615826984181789921790, 4.62251208921023152475242741351, 6.04347984886334806470054103310, 6.28165334955427960098520591649, 7.02657822281629818148146991200, 8.627913734733030451301225583250, 9.280162239023777035467651422160, 10.06075186511197498786992640977
