| L(s) = 1 | − 3-s + 5-s + 9-s − 2·11-s − 4·13-s − 15-s + 17-s − 7·19-s − 5·23-s − 4·25-s − 27-s − 2·29-s + 8·31-s + 2·33-s + 11·37-s + 4·39-s + 12·41-s − 5·43-s + 45-s + 6·47-s − 51-s − 2·53-s − 2·55-s + 7·57-s + 5·59-s + 2·61-s − 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.258·15-s + 0.242·17-s − 1.60·19-s − 1.04·23-s − 4/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.348·33-s + 1.80·37-s + 0.640·39-s + 1.87·41-s − 0.762·43-s + 0.149·45-s + 0.875·47-s − 0.140·51-s − 0.274·53-s − 0.269·55-s + 0.927·57-s + 0.650·59-s + 0.256·61-s − 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8017032736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8017032736\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.14992747385132, −12.79903475916306, −12.38346312109850, −11.85776192804590, −11.39000337532608, −10.92349238658265, −10.31286525609946, −9.952761201085824, −9.693482997531537, −9.052846701088000, −8.267276917480683, −8.026682496303691, −7.421017245158203, −6.925680270611670, −6.240284972634654, −5.892903508091799, −5.537586897015240, −4.761522148802504, −4.261766362945515, −4.029352165026079, −2.851449568325373, −2.468638220119917, −1.996643165199237, −1.131547582056418, −0.2824046262043159,
0.2824046262043159, 1.131547582056418, 1.996643165199237, 2.468638220119917, 2.851449568325373, 4.029352165026079, 4.261766362945515, 4.761522148802504, 5.537586897015240, 5.892903508091799, 6.240284972634654, 6.925680270611670, 7.421017245158203, 8.026682496303691, 8.267276917480683, 9.052846701088000, 9.693482997531537, 9.952761201085824, 10.31286525609946, 10.92349238658265, 11.39000337532608, 11.85776192804590, 12.38346312109850, 12.79903475916306, 13.14992747385132
