| L(s) = 1 | + 2.71·2-s + 5.39·4-s + 3.51·5-s − 3.83·7-s + 9.22·8-s + 9.54·10-s − 1.15·13-s − 10.4·14-s + 14.3·16-s + 5.40·17-s + 0.558·19-s + 18.9·20-s − 2.75·23-s + 7.32·25-s − 3.12·26-s − 20.6·28-s − 4.70·29-s + 8.76·31-s + 20.4·32-s + 14.6·34-s − 13.4·35-s − 4.31·37-s + 1.51·38-s + 32.3·40-s + 0.476·41-s + 3.24·43-s − 7.48·46-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.69·4-s + 1.56·5-s − 1.44·7-s + 3.26·8-s + 3.01·10-s − 0.319·13-s − 2.78·14-s + 3.57·16-s + 1.31·17-s + 0.128·19-s + 4.23·20-s − 0.574·23-s + 1.46·25-s − 0.613·26-s − 3.90·28-s − 0.874·29-s + 1.57·31-s + 3.61·32-s + 2.52·34-s − 2.27·35-s − 0.709·37-s + 0.246·38-s + 5.12·40-s + 0.0743·41-s + 0.494·43-s − 1.10·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.951835269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.951835269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 13 | \( 1 + 1.15T + 13T^{2} \) |
| 17 | \( 1 - 5.40T + 17T^{2} \) |
| 19 | \( 1 - 0.558T + 19T^{2} \) |
| 23 | \( 1 + 2.75T + 23T^{2} \) |
| 29 | \( 1 + 4.70T + 29T^{2} \) |
| 31 | \( 1 - 8.76T + 31T^{2} \) |
| 37 | \( 1 + 4.31T + 37T^{2} \) |
| 41 | \( 1 - 0.476T + 41T^{2} \) |
| 43 | \( 1 - 3.24T + 43T^{2} \) |
| 47 | \( 1 - 4.42T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 6.79T + 59T^{2} \) |
| 61 | \( 1 + 5.53T + 61T^{2} \) |
| 67 | \( 1 - 6.70T + 67T^{2} \) |
| 71 | \( 1 - 5.88T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.12527306787632070853049163389, −6.69762775099621185048652192472, −5.99787886799053135570753414126, −5.66559694426020150486267327587, −5.15221195108629837199008237344, −4.12464336872101141780216469509, −3.45796282069587645194193429798, −2.73088817251555242432180744253, −2.25163335850922367096809829080, −1.18792169670792347113174024706,
1.18792169670792347113174024706, 2.25163335850922367096809829080, 2.73088817251555242432180744253, 3.45796282069587645194193429798, 4.12464336872101141780216469509, 5.15221195108629837199008237344, 5.66559694426020150486267327587, 5.99787886799053135570753414126, 6.69762775099621185048652192472, 7.12527306787632070853049163389
