| L(s) = 1 | − 2-s + 4-s + 0.732·5-s + 7-s − 8-s − 0.732·10-s − 3.46·11-s + 5.46·13-s − 14-s + 16-s − 2.73·17-s − 2·19-s + 0.732·20-s + 3.46·22-s − 23-s − 4.46·25-s − 5.46·26-s + 28-s − 8·29-s − 10.1·31-s − 32-s + 2.73·34-s + 0.732·35-s + 7.46·37-s + 2·38-s − 0.732·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.327·5-s + 0.377·7-s − 0.353·8-s − 0.231·10-s − 1.04·11-s + 1.51·13-s − 0.267·14-s + 0.250·16-s − 0.662·17-s − 0.458·19-s + 0.163·20-s + 0.738·22-s − 0.208·23-s − 0.892·25-s − 1.07·26-s + 0.188·28-s − 1.48·29-s − 1.83·31-s − 0.176·32-s + 0.468·34-s + 0.123·35-s + 1.22·37-s + 0.324·38-s − 0.115·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 2.73T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 + 5.66T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 0.535T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 + 3.80T + 89T^{2} \) |
| 97 | \( 1 + 5.26T + 97T^{2} \) |
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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
−8.419844399867227504734333844765, −7.76153100644054777353324239760, −7.07562304732909573071995871292, −5.92903874395923770611690307388, −5.69036877399604018393798690692, −4.39497980279158690824289828256, −3.49535244114461844856153563601, −2.30385092076484685652096088854, −1.54777570597275483446057107967, 0,
1.54777570597275483446057107967, 2.30385092076484685652096088854, 3.49535244114461844856153563601, 4.39497980279158690824289828256, 5.69036877399604018393798690692, 5.92903874395923770611690307388, 7.07562304732909573071995871292, 7.76153100644054777353324239760, 8.419844399867227504734333844765
