| L(s) = 1 | + 1.07·2-s − 2.43·3-s − 0.848·4-s − 0.625·5-s − 2.60·6-s − 3.05·8-s + 2.91·9-s − 0.671·10-s + 0.708·11-s + 2.06·12-s + 1.52·15-s − 1.58·16-s − 3.34·17-s + 3.12·18-s + 5.20·19-s + 0.530·20-s + 0.760·22-s − 4.43·23-s + 7.43·24-s − 4.60·25-s + 0.214·27-s − 6.59·29-s + 1.63·30-s + 4.39·31-s + 4.41·32-s − 1.72·33-s − 3.58·34-s + ⋯ |
| L(s) = 1 | + 0.758·2-s − 1.40·3-s − 0.424·4-s − 0.279·5-s − 1.06·6-s − 1.08·8-s + 0.970·9-s − 0.212·10-s + 0.213·11-s + 0.595·12-s + 0.392·15-s − 0.395·16-s − 0.810·17-s + 0.736·18-s + 1.19·19-s + 0.118·20-s + 0.162·22-s − 0.924·23-s + 1.51·24-s − 0.921·25-s + 0.0413·27-s − 1.22·29-s + 0.297·30-s + 0.788·31-s + 0.780·32-s − 0.299·33-s − 0.615·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5545247527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5545247527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.07T + 2T^{2} \) |
| 3 | \( 1 + 2.43T + 3T^{2} \) |
| 5 | \( 1 + 0.625T + 5T^{2} \) |
| 11 | \( 1 - 0.708T + 11T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 - 5.20T + 19T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 + 6.59T + 29T^{2} \) |
| 31 | \( 1 - 4.39T + 31T^{2} \) |
| 37 | \( 1 + 0.423T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.07T + 47T^{2} \) |
| 53 | \( 1 + 0.697T + 53T^{2} \) |
| 59 | \( 1 + 9.86T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 + 3.91T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 0.202T + 97T^{2} \) |
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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
−7.74091029255161349921498350580, −6.72684504006284515818350821345, −6.29186640177507725863859149073, −5.54411110315149792017819210427, −5.14731007129038438760167655017, −4.37088301616175694168205819117, −3.82441476908038242117414877436, −2.92463746777287656996717238056, −1.61267530052584760397806687908, −0.35400359132230356953452010474,
0.35400359132230356953452010474, 1.61267530052584760397806687908, 2.92463746777287656996717238056, 3.82441476908038242117414877436, 4.37088301616175694168205819117, 5.14731007129038438760167655017, 5.54411110315149792017819210427, 6.29186640177507725863859149073, 6.72684504006284515818350821345, 7.74091029255161349921498350580
