| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4478623668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4478623668\) |
| \(L(2)\) |
|
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 | \( 1 - 128.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00070983171187328373971546160, −9.157941678725945322785516245829, −8.153738732771968324846178964729, −7.43732343023138919393235368342, −6.64558903196124944357253649059, −5.65921717136614753132951468959, −4.37428211918763423502599801153, −3.47638981803897568788029708312, −2.74779738443407932133926357637, −0.40145765240508585858785691284,
0.40145765240508585858785691284, 2.74779738443407932133926357637, 3.47638981803897568788029708312, 4.37428211918763423502599801153, 5.65921717136614753132951468959, 6.64558903196124944357253649059, 7.43732343023138919393235368342, 8.153738732771968324846178964729, 9.157941678725945322785516245829, 10.00070983171187328373971546160
