| L(s) = 1 | + 2.69·2-s + 5.28·4-s − 2.77·5-s + 1.18·7-s + 8.85·8-s − 7.49·10-s − 11-s − 6.75·13-s + 3.20·14-s + 13.3·16-s − 3.55·17-s − 7.48·19-s − 14.6·20-s − 2.69·22-s + 7.70·23-s + 2.72·25-s − 18.2·26-s + 6.27·28-s − 4.03·29-s + 8.18·31-s + 18.2·32-s − 9.58·34-s − 3.30·35-s + 3.73·37-s − 20.1·38-s − 24.6·40-s − 2.16·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.64·4-s − 1.24·5-s + 0.449·7-s + 3.13·8-s − 2.37·10-s − 0.301·11-s − 1.87·13-s + 0.856·14-s + 3.33·16-s − 0.861·17-s − 1.71·19-s − 3.28·20-s − 0.575·22-s + 1.60·23-s + 0.544·25-s − 3.57·26-s + 1.18·28-s − 0.749·29-s + 1.46·31-s + 3.23·32-s − 1.64·34-s − 0.558·35-s + 0.614·37-s − 3.27·38-s − 3.89·40-s − 0.337·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 1.18T + 7T^{2} \) |
| 13 | \( 1 + 6.75T + 13T^{2} \) |
| 17 | \( 1 + 3.55T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 - 3.73T + 37T^{2} \) |
| 41 | \( 1 + 2.16T + 41T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 + 0.681T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 1.40T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 - 1.76T + 97T^{2} \) |
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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
−7.16273024759931938043171165955, −6.80671758976647915997141936023, −5.99806298001326770459710714910, −4.84264752630971624838071449739, −4.75850633675900942991406935270, −4.20933344718379031872195489799, −3.19064821440356627708011295865, −2.63507900409441816623319178486, −1.78816125774905502245941576908, 0,
1.78816125774905502245941576908, 2.63507900409441816623319178486, 3.19064821440356627708011295865, 4.20933344718379031872195489799, 4.75850633675900942991406935270, 4.84264752630971624838071449739, 5.99806298001326770459710714910, 6.80671758976647915997141936023, 7.16273024759931938043171165955
