| L(s) = 1 | − 15.4·5-s + 23.8·7-s + 14.2·11-s + 13.8·13-s + 80.5·17-s + 144.·19-s + 141.·23-s + 112.·25-s − 251.·29-s − 16.6·31-s − 367.·35-s − 305.·37-s − 429.·41-s + 181.·43-s + 79.4·47-s + 225.·49-s − 663.·53-s − 219.·55-s + 220.·59-s + 473.·61-s − 213.·65-s + 647.·67-s − 14.4·71-s + 776.·73-s + 339.·77-s − 257.·79-s + 1.28e3·83-s + ⋯ |
| L(s) = 1 | − 1.37·5-s + 1.28·7-s + 0.390·11-s + 0.295·13-s + 1.14·17-s + 1.74·19-s + 1.27·23-s + 0.901·25-s − 1.60·29-s − 0.0965·31-s − 1.77·35-s − 1.35·37-s − 1.63·41-s + 0.644·43-s + 0.246·47-s + 0.655·49-s − 1.72·53-s − 0.538·55-s + 0.486·59-s + 0.993·61-s − 0.406·65-s + 1.18·67-s − 0.0242·71-s + 1.24·73-s + 0.502·77-s − 0.367·79-s + 1.69·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.208113255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208113255\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 - 23.8T + 343T^{2} \) |
| 11 | \( 1 - 14.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 80.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 251.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 16.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 79.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 220.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 473.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 647.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 14.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 257.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 156.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−8.824175350331989322194491701653, −7.962329316961856912054696509992, −7.59964695747235742277194189476, −6.84696376979825856880117968712, −5.35632669414955620631450574763, −5.00070823066020826338269587984, −3.75593557045953240798197042368, −3.31639389024373499978039220770, −1.65376343970714549612173030762, −0.75841012413738186725470397206,
0.75841012413738186725470397206, 1.65376343970714549612173030762, 3.31639389024373499978039220770, 3.75593557045953240798197042368, 5.00070823066020826338269587984, 5.35632669414955620631450574763, 6.84696376979825856880117968712, 7.59964695747235742277194189476, 7.962329316961856912054696509992, 8.824175350331989322194491701653
