| L(s) = 1 | − 2-s + 1.63·3-s + 4-s − 2.11·5-s − 1.63·6-s − 4.12·7-s − 8-s − 0.321·9-s + 2.11·10-s − 3.16·11-s + 1.63·12-s − 4.77·13-s + 4.12·14-s − 3.45·15-s + 16-s + 1.92·17-s + 0.321·18-s − 6.77·19-s − 2.11·20-s − 6.74·21-s + 3.16·22-s − 7.80·23-s − 1.63·24-s − 0.537·25-s + 4.77·26-s − 5.43·27-s − 4.12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.944·3-s + 0.5·4-s − 0.944·5-s − 0.668·6-s − 1.55·7-s − 0.353·8-s − 0.107·9-s + 0.668·10-s − 0.955·11-s + 0.472·12-s − 1.32·13-s + 1.10·14-s − 0.892·15-s + 0.250·16-s + 0.467·17-s + 0.0756·18-s − 1.55·19-s − 0.472·20-s − 1.47·21-s + 0.675·22-s − 1.62·23-s − 0.334·24-s − 0.107·25-s + 0.936·26-s − 1.04·27-s − 0.779·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3315033258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3315033258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + 4.77T + 13T^{2} \) |
| 17 | \( 1 - 1.92T + 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 23 | \( 1 + 7.80T + 23T^{2} \) |
| 29 | \( 1 - 9.35T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 8.13T + 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 + 8.16T + 59T^{2} \) |
| 61 | \( 1 - 3.29T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 8.90T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−8.279608121891284386425748066709, −7.935001008870046716727947026232, −7.26894181789831343106493875614, −6.43421570888384951948556767109, −5.69513397776040372989633535216, −4.36373533861289926455235005503, −3.65438984203740916871320559699, −2.64141248656499139368525356962, −2.43132792006063906196099235036, −0.31576876403342693090998556663,
0.31576876403342693090998556663, 2.43132792006063906196099235036, 2.64141248656499139368525356962, 3.65438984203740916871320559699, 4.36373533861289926455235005503, 5.69513397776040372989633535216, 6.43421570888384951948556767109, 7.26894181789831343106493875614, 7.935001008870046716727947026232, 8.279608121891284386425748066709
