| L(s) = 1 | + 1.38·2-s − 0.0779·4-s − 2.26·5-s − 0.965·7-s − 2.88·8-s − 3.14·10-s − 11-s − 0.729·13-s − 1.33·14-s − 3.83·16-s − 2.76·17-s − 1.50·19-s + 0.176·20-s − 1.38·22-s − 0.724·23-s + 0.143·25-s − 1.01·26-s + 0.0752·28-s − 7.22·29-s − 9.30·31-s + 0.440·32-s − 3.83·34-s + 2.18·35-s + 10.5·37-s − 2.08·38-s + 6.53·40-s + 11.0·41-s + ⋯ |
| L(s) = 1 | + 0.980·2-s − 0.0389·4-s − 1.01·5-s − 0.364·7-s − 1.01·8-s − 0.994·10-s − 0.301·11-s − 0.202·13-s − 0.357·14-s − 0.959·16-s − 0.671·17-s − 0.344·19-s + 0.0395·20-s − 0.295·22-s − 0.151·23-s + 0.0286·25-s − 0.198·26-s + 0.0142·28-s − 1.34·29-s − 1.67·31-s + 0.0778·32-s − 0.658·34-s + 0.370·35-s + 1.74·37-s − 0.337·38-s + 1.03·40-s + 1.73·41-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
| \(L(1)\) |
\(\approx\) |
\(0.8977845090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8977845090\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.38T + 2T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 + 0.965T + 7T^{2} \) |
| 13 | \( 1 + 0.729T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 1.50T + 19T^{2} \) |
| 23 | \( 1 + 0.724T + 23T^{2} \) |
| 29 | \( 1 + 7.22T + 29T^{2} \) |
| 31 | \( 1 + 9.30T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 - 2.01T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6.02T + 61T^{2} \) |
| 67 | \( 1 + 7.37T + 67T^{2} \) |
| 71 | \( 1 + 8.07T + 71T^{2} \) |
| 73 | \( 1 - 0.436T + 73T^{2} \) |
| 79 | \( 1 + 2.79T + 79T^{2} \) |
| 83 | \( 1 + 3.20T + 83T^{2} \) |
| 89 | \( 1 - 5.27T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.56716092754669559610870282861, −7.29046153699223951981949484479, −6.09546963719080185652377096847, −5.84247935758281275576618059680, −4.79413755928274821343241994868, −4.27438622154001770191980352826, −3.69844328836078098442844457290, −2.98734597627999283889093440396, −2.06246630244491140300992409805, −0.38037017388995179752260862732,
0.38037017388995179752260862732, 2.06246630244491140300992409805, 2.98734597627999283889093440396, 3.69844328836078098442844457290, 4.27438622154001770191980352826, 4.79413755928274821343241994868, 5.84247935758281275576618059680, 6.09546963719080185652377096847, 7.29046153699223951981949484479, 7.56716092754669559610870282861
