| L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s + 2·13-s + 15-s − 4·17-s + 23-s + 25-s − 27-s + 29-s + 2·31-s + 2·33-s + 2·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s − 7·49-s + 4·51-s − 10·53-s + 2·55-s + 10·59-s − 2·61-s − 2·65-s − 69-s − 10·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 0.258·15-s − 0.970·17-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.359·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s + 0.560·51-s − 1.37·53-s + 0.269·55-s + 1.30·59-s − 0.256·61-s − 0.248·65-s − 0.120·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085982640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085982640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.24275746408438, −12.82783162433102, −12.32961464357928, −11.77448076517176, −11.35671794144766, −10.89651661548649, −10.62943706274777, −10.01518446255552, −9.425287397154051, −8.989068539575669, −8.390311064602453, −7.938559538309436, −7.465718456473718, −6.888577727672710, −6.352923839260818, −6.002392411299102, −5.303661091613818, −4.817688042295810, −4.306834106408254, −3.844491971030345, −3.061764544310201, −2.578056861533233, −1.808844267687734, −1.078047425602228, −0.3523202827421212,
0.3523202827421212, 1.078047425602228, 1.808844267687734, 2.578056861533233, 3.061764544310201, 3.844491971030345, 4.306834106408254, 4.817688042295810, 5.303661091613818, 6.002392411299102, 6.352923839260818, 6.888577727672710, 7.465718456473718, 7.938559538309436, 8.390311064602453, 8.989068539575669, 9.425287397154051, 10.01518446255552, 10.62943706274777, 10.89651661548649, 11.35671794144766, 11.77448076517176, 12.32961464357928, 12.82783162433102, 13.24275746408438
