| L(s) = 1 | − 2-s + 4-s − 0.585·5-s − 1.49·7-s − 8-s + 0.585·10-s − 3.18·11-s − 6.04·13-s + 1.49·14-s + 16-s + 4.54·17-s + 4.74·19-s − 0.585·20-s + 3.18·22-s + 1.06·23-s − 4.65·25-s + 6.04·26-s − 1.49·28-s + 10.4·29-s − 6.42·31-s − 32-s − 4.54·34-s + 0.876·35-s − 9.98·37-s − 4.74·38-s + 0.585·40-s + 3.37·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.261·5-s − 0.565·7-s − 0.353·8-s + 0.185·10-s − 0.958·11-s − 1.67·13-s + 0.400·14-s + 0.250·16-s + 1.10·17-s + 1.08·19-s − 0.130·20-s + 0.678·22-s + 0.222·23-s − 0.931·25-s + 1.18·26-s − 0.282·28-s + 1.93·29-s − 1.15·31-s − 0.176·32-s − 0.779·34-s + 0.148·35-s − 1.64·37-s − 0.770·38-s + 0.0926·40-s + 0.527·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7214783487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7214783487\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 - 1.06T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + 9.98T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + 7.71T + 53T^{2} \) |
| 59 | \( 1 + 3.21T + 59T^{2} \) |
| 61 | \( 1 - 5.85T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 4.50T + 79T^{2} \) |
| 83 | \( 1 - 1.09T + 83T^{2} \) |
| 89 | \( 1 - 4.43T + 89T^{2} \) |
| 97 | \( 1 + 7.60T + 97T^{2} \) |
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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
−8.165125195424258051241262868912, −7.890730124658814377635734086948, −7.12246503426262409221743781421, −6.49808620088286618359378534769, −5.28338523995771295301050988924, −5.02175548683418308914229521902, −3.51146829269541668260443619705, −2.93582087930339468492685138405, −1.93914870603288113416770632022, −0.51766169712751297155567303387,
0.51766169712751297155567303387, 1.93914870603288113416770632022, 2.93582087930339468492685138405, 3.51146829269541668260443619705, 5.02175548683418308914229521902, 5.28338523995771295301050988924, 6.49808620088286618359378534769, 7.12246503426262409221743781421, 7.890730124658814377635734086948, 8.165125195424258051241262868912
