| L(s) = 1 | + 2-s − 1.35·3-s + 4-s − 5-s − 1.35·6-s + 8-s − 1.17·9-s − 10-s − 2.93·11-s − 1.35·12-s + 4.16·13-s + 1.35·15-s + 16-s − 17-s − 1.17·18-s − 2.23·19-s − 20-s − 2.93·22-s + 8.69·23-s − 1.35·24-s + 25-s + 4.16·26-s + 5.64·27-s + 2.53·29-s + 1.35·30-s − 9.15·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.780·3-s + 0.5·4-s − 0.447·5-s − 0.552·6-s + 0.353·8-s − 0.390·9-s − 0.316·10-s − 0.886·11-s − 0.390·12-s + 1.15·13-s + 0.349·15-s + 0.250·16-s − 0.242·17-s − 0.276·18-s − 0.511·19-s − 0.223·20-s − 0.626·22-s + 1.81·23-s − 0.276·24-s + 0.200·25-s + 0.816·26-s + 1.08·27-s + 0.470·29-s + 0.246·30-s − 1.64·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.714865040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.714865040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 8.69T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 + 9.15T + 31T^{2} \) |
| 37 | \( 1 + 0.103T + 37T^{2} \) |
| 41 | \( 1 + 4.99T + 41T^{2} \) |
| 43 | \( 1 - 0.552T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 + 0.109T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 - 6.82T + 61T^{2} \) |
| 67 | \( 1 - 3.16T + 67T^{2} \) |
| 71 | \( 1 - 1.49T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 + 6.69T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 - 0.714T + 97T^{2} \) |
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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
−7.64308613887293886984414116861, −6.88966524951070900561760804252, −6.35956975407520625922289085747, −5.58857817978569278806217261501, −5.09660391046139249219269733173, −4.44661682442632718680647269326, −3.44253350548832584140818117964, −2.95677135976353422558835390975, −1.79034892258982050589240464390, −0.59252730779496877820146982347,
0.59252730779496877820146982347, 1.79034892258982050589240464390, 2.95677135976353422558835390975, 3.44253350548832584140818117964, 4.44661682442632718680647269326, 5.09660391046139249219269733173, 5.58857817978569278806217261501, 6.35956975407520625922289085747, 6.88966524951070900561760804252, 7.64308613887293886984414116861
