L(s) = 1 | + 2.73·2-s + 5.46·4-s + 5-s + 9.46·8-s + 2.73·10-s − 0.732·11-s − 2.26·13-s + 14.9·16-s + 3.26·17-s − 4.46·19-s + 5.46·20-s − 2·22-s + 4.73·23-s + 25-s − 6.19·26-s + 4.19·29-s + 0.464·31-s + 21.8·32-s + 8.92·34-s − 3.19·37-s − 12.1·38-s + 9.46·40-s − 0.732·41-s + 3.19·43-s − 4·44-s + 12.9·46-s + 2·47-s + ⋯ |
L(s) = 1 | + 1.93·2-s + 2.73·4-s + 0.447·5-s + 3.34·8-s + 0.863·10-s − 0.220·11-s − 0.629·13-s + 3.73·16-s + 0.792·17-s − 1.02·19-s + 1.22·20-s − 0.426·22-s + 0.986·23-s + 0.200·25-s − 1.21·26-s + 0.779·29-s + 0.0833·31-s + 3.86·32-s + 1.53·34-s − 0.525·37-s − 1.97·38-s + 1.49·40-s − 0.114·41-s + 0.487·43-s − 0.603·44-s + 1.90·46-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.793077856\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.793077856\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.73T + 2T^{2} \) |
| 11 | \( 1 + 0.732T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 0.464T + 31T^{2} \) |
| 37 | \( 1 + 3.19T + 37T^{2} \) |
| 41 | \( 1 + 0.732T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 0.196T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 + 6.19T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 7.39T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−9.070815285831849735504544354741, −7.901649637049607527798861593735, −7.18116904729396565300064076755, −6.41064938264516690572384493412, −5.76823206927625251967800913820, −4.94303185529883518136152078696, −4.40789050424413627140145966940, −3.26233120230207922111087899340, −2.62354844400724227469842370311, −1.56034307809233887657442766841,
1.56034307809233887657442766841, 2.62354844400724227469842370311, 3.26233120230207922111087899340, 4.40789050424413627140145966940, 4.94303185529883518136152078696, 5.76823206927625251967800913820, 6.41064938264516690572384493412, 7.18116904729396565300064076755, 7.901649637049607527798861593735, 9.070815285831849735504544354741