| L(s) = 1 | + 0.368·2-s + 3-s − 1.86·4-s − 0.572·5-s + 0.368·6-s − 1.42·8-s + 9-s − 0.210·10-s + 5.73·11-s − 1.86·12-s − 3.53·13-s − 0.572·15-s + 3.20·16-s − 1.83·17-s + 0.368·18-s − 5.95·19-s + 1.06·20-s + 2.11·22-s + 3.12·23-s − 1.42·24-s − 4.67·25-s − 1.30·26-s + 27-s − 0.0143·29-s − 0.210·30-s + 6.00·31-s + 4.02·32-s + ⋯ |
| L(s) = 1 | + 0.260·2-s + 0.577·3-s − 0.932·4-s − 0.255·5-s + 0.150·6-s − 0.502·8-s + 0.333·9-s − 0.0666·10-s + 1.72·11-s − 0.538·12-s − 0.981·13-s − 0.147·15-s + 0.801·16-s − 0.444·17-s + 0.0867·18-s − 1.36·19-s + 0.238·20-s + 0.449·22-s + 0.650·23-s − 0.290·24-s − 0.934·25-s − 0.255·26-s + 0.192·27-s − 0.00266·29-s − 0.0384·30-s + 1.07·31-s + 0.711·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.948794147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.948794147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.368T + 2T^{2} \) |
| 5 | \( 1 + 0.572T + 5T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 0.0143T + 29T^{2} \) |
| 31 | \( 1 - 6.00T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 43 | \( 1 - 6.01T + 43T^{2} \) |
| 47 | \( 1 + 6.07T + 47T^{2} \) |
| 53 | \( 1 + 0.555T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 + 8.02T + 73T^{2} \) |
| 79 | \( 1 - 5.75T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 2.84T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.189702599911110562114699432287, −7.47565047964836720389242266470, −6.55792975542682659754022788219, −6.08034173744472020470896197062, −4.88009883409803645395015781830, −4.32688944983247045381743423526, −3.86837907737070287695445743055, −2.92449079022441417090969239876, −1.91957676071327498224224363829, −0.69314298251849407574937718834,
0.69314298251849407574937718834, 1.91957676071327498224224363829, 2.92449079022441417090969239876, 3.86837907737070287695445743055, 4.32688944983247045381743423526, 4.88009883409803645395015781830, 6.08034173744472020470896197062, 6.55792975542682659754022788219, 7.47565047964836720389242266470, 8.189702599911110562114699432287
