| L(s) = 1 | − 9.48·5-s + 3.48·7-s − 21.9·11-s + 64.9·25-s + 50·29-s − 23.4·31-s − 33.0·35-s − 36.8·49-s + 89.4·53-s + 208.·55-s + 10·59-s + 93.7·73-s − 76.5·77-s + 58·79-s + 151.·83-s + 61.0·97-s + 35.6·101-s + 10·103-s − 126.·107-s + ⋯ |
| L(s) = 1 | − 1.89·5-s + 0.497·7-s − 1.99·11-s + 2.59·25-s + 1.72·29-s − 0.755·31-s − 0.944·35-s − 0.752·49-s + 1.68·53-s + 3.78·55-s + 0.169·59-s + 1.28·73-s − 0.994·77-s + 0.734·79-s + 1.82·83-s + 0.629·97-s + 0.352·101-s + 0.0970·103-s − 1.18·107-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8747665152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8747665152\) |
| \(L(2)\) |
|
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 | \( 1 + 9.48T + 25T^{2} \) |
| 7 | \( 1 - 3.48T + 49T^{2} \) |
| 11 | \( 1 + 21.9T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 50T + 841T^{2} \) |
| 31 | \( 1 + 23.4T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 89.4T + 2.80e3T^{2} \) |
| 59 | \( 1 - 10T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 93.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58T + 6.24e3T^{2} \) |
| 83 | \( 1 - 151.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−10.27698690499847492894371733495, −8.817320071468195396354310990645, −8.011126648380145098618843864587, −7.74577637533978357897971955422, −6.77168657627613631206641639461, −5.26021288921134958447220360003, −4.63818243729398263996721538850, −3.57019412820927212081875870152, −2.56396859392918943898518990170, −0.57864916617339310231452718149,
0.57864916617339310231452718149, 2.56396859392918943898518990170, 3.57019412820927212081875870152, 4.63818243729398263996721538850, 5.26021288921134958447220360003, 6.77168657627613631206641639461, 7.74577637533978357897971955422, 8.011126648380145098618843864587, 8.817320071468195396354310990645, 10.27698690499847492894371733495
