| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s + 9-s − 4·11-s + 13-s − 4·15-s − 17-s − 4·19-s + 4·21-s + 4·23-s − 25-s − 4·27-s + 2·31-s − 8·33-s − 4·35-s − 2·37-s + 2·39-s − 4·41-s + 4·43-s − 2·45-s − 3·49-s − 2·51-s + 6·53-s + 8·55-s − 8·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 1.03·15-s − 0.242·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.359·31-s − 1.39·33-s − 0.676·35-s − 0.328·37-s + 0.320·39-s − 0.624·41-s + 0.609·43-s − 0.298·45-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 1.07·55-s − 1.05·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.238014138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238014138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.92630169529500, −15.32286880883047, −15.14721573834680, −14.55713476757205, −13.94754760678869, −13.34267920403573, −12.98019208580872, −12.20257868184523, −11.56430227974746, −11.00186320870209, −10.54059149609178, −9.779802404864237, −8.988686241446292, −8.427020458197504, −8.161714311226792, −7.585514264660792, −7.026898532652670, −6.090302834943587, −5.206089692582305, −4.650547343434093, −3.857020600323570, −3.306142302185296, −2.459341449348559, −1.936360149559293, −0.5956604024355093,
0.5956604024355093, 1.936360149559293, 2.459341449348559, 3.306142302185296, 3.857020600323570, 4.650547343434093, 5.206089692582305, 6.090302834943587, 7.026898532652670, 7.585514264660792, 8.161714311226792, 8.427020458197504, 8.988686241446292, 9.779802404864237, 10.54059149609178, 11.00186320870209, 11.56430227974746, 12.20257868184523, 12.98019208580872, 13.34267920403573, 13.94754760678869, 14.55713476757205, 15.14721573834680, 15.32286880883047, 15.92630169529500
