| L(s) = 1 | − 8.27·5-s + 8.82·7-s + 9·9-s + 17.3·11-s − 33.9·17-s − 19·19-s + 30·23-s + 43.4·25-s − 73.0·35-s + 31.1·43-s − 74.4·45-s − 11.5·47-s + 28.8·49-s − 143.·55-s + 108.·61-s + 79.4·63-s + 137.·73-s + 153.·77-s + 81·81-s + 90·83-s + 280.·85-s + 157.·95-s + 156.·99-s + 102·101-s − 248.·115-s − 299.·119-s + ⋯ |
| L(s) = 1 | − 1.65·5-s + 1.26·7-s + 9-s + 1.57·11-s − 1.99·17-s − 19-s + 1.30·23-s + 1.73·25-s − 2.08·35-s + 0.725·43-s − 1.65·45-s − 0.246·47-s + 0.589·49-s − 2.61·55-s + 1.77·61-s + 1.26·63-s + 1.87·73-s + 1.99·77-s + 81-s + 1.08·83-s + 3.30·85-s + 1.65·95-s + 1.57·99-s + 1.00·101-s − 2.15·115-s − 2.51·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.826417739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.826417739\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 5 | \( 1 + 8.27T + 25T^{2} \) |
| 7 | \( 1 - 8.82T + 49T^{2} \) |
| 11 | \( 1 - 17.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 33.9T + 289T^{2} \) |
| 23 | \( 1 - 30T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 31.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 90T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.249072203508443292901674703241, −8.663909534111799877263841375591, −7.959378767688993249294202391933, −7.00112076372468266461236978756, −6.63048977005336961018261782124, −4.82075029923866778757549980192, −4.33692293712700051648747847624, −3.74204118672950998808912107182, −2.03830946800499647329407192931, −0.840945592279295720297140119457,
0.840945592279295720297140119457, 2.03830946800499647329407192931, 3.74204118672950998808912107182, 4.33692293712700051648747847624, 4.82075029923866778757549980192, 6.63048977005336961018261782124, 7.00112076372468266461236978756, 7.959378767688993249294202391933, 8.663909534111799877263841375591, 9.249072203508443292901674703241
