| L(s) = 1 | + 5·5-s + 20·7-s − 16·11-s + 58·13-s − 38·17-s + 4·19-s + 80·23-s + 25·25-s − 82·29-s − 8·31-s + 100·35-s + 426·37-s + 246·41-s − 524·43-s + 464·47-s + 57·49-s + 702·53-s − 80·55-s + 592·59-s + 574·61-s + 290·65-s − 172·67-s − 768·71-s − 558·73-s − 320·77-s + 408·79-s − 164·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.07·7-s − 0.438·11-s + 1.23·13-s − 0.542·17-s + 0.0482·19-s + 0.725·23-s + 1/5·25-s − 0.525·29-s − 0.0463·31-s + 0.482·35-s + 1.89·37-s + 0.937·41-s − 1.85·43-s + 1.44·47-s + 0.166·49-s + 1.81·53-s − 0.196·55-s + 1.30·59-s + 1.20·61-s + 0.553·65-s − 0.313·67-s − 1.28·71-s − 0.894·73-s − 0.473·77-s + 0.581·79-s − 0.216·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.393772897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393772897\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 80 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 8 T + p^{3} T^{2} \) |
| 37 | \( 1 - 426 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 524 T + p^{3} T^{2} \) |
| 47 | \( 1 - 464 T + p^{3} T^{2} \) |
| 53 | \( 1 - 702 T + p^{3} T^{2} \) |
| 59 | \( 1 - 592 T + p^{3} T^{2} \) |
| 61 | \( 1 - 574 T + p^{3} T^{2} \) |
| 67 | \( 1 + 172 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 558 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 + 164 T + p^{3} T^{2} \) |
| 89 | \( 1 - 510 T + p^{3} T^{2} \) |
| 97 | \( 1 - 514 T + p^{3} T^{2} \) |
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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
−11.08639269691535931871650424562, −10.22757289117123376173863843251, −9.020155785402700106552813916232, −8.317538363022964981600854385717, −7.28080977256345250374505617433, −6.07595096726552326967379925018, −5.14773570277861915469857679848, −4.01078851087135331060282197433, −2.43837740220450195187372195228, −1.13166410536671084863059251183,
1.13166410536671084863059251183, 2.43837740220450195187372195228, 4.01078851087135331060282197433, 5.14773570277861915469857679848, 6.07595096726552326967379925018, 7.28080977256345250374505617433, 8.317538363022964981600854385717, 9.020155785402700106552813916232, 10.22757289117123376173863843251, 11.08639269691535931871650424562
