L(s) = 1 | − 2.61·2-s + 1.26·3-s + 4.85·4-s + 0.306·5-s − 3.30·6-s − 1.43·7-s − 7.47·8-s − 1.41·9-s − 0.802·10-s + 2.14·11-s + 6.11·12-s − 3.19·13-s + 3.76·14-s + 0.386·15-s + 9.85·16-s − 3.80·17-s + 3.69·18-s + 4.63·19-s + 1.48·20-s − 1.81·21-s − 5.61·22-s − 3.20·23-s − 9.41·24-s − 4.90·25-s + 8.37·26-s − 5.56·27-s − 6.98·28-s + ⋯ |
L(s) = 1 | − 1.85·2-s + 0.727·3-s + 2.42·4-s + 0.137·5-s − 1.34·6-s − 0.543·7-s − 2.64·8-s − 0.470·9-s − 0.253·10-s + 0.646·11-s + 1.76·12-s − 0.887·13-s + 1.00·14-s + 0.0997·15-s + 2.46·16-s − 0.923·17-s + 0.870·18-s + 1.06·19-s + 0.332·20-s − 0.395·21-s − 1.19·22-s − 0.668·23-s − 1.92·24-s − 0.981·25-s + 1.64·26-s − 1.07·27-s − 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5771997357\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5771997357\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 8011 | \( 1+O(T) \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 1.26T + 3T^{2} \) |
| 5 | \( 1 - 0.306T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 - 2.14T + 11T^{2} \) |
| 13 | \( 1 + 3.19T + 13T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 + 3.20T + 23T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 + 3.98T + 31T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 + 7.51T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 + 1.13T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 1.57T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 0.359T + 73T^{2} \) |
| 79 | \( 1 + 0.646T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 + 8.54T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.044209681276593803743468800061, −7.39523340890891233347288630205, −6.77795789414029725958832711226, −6.12335033972654218210746491807, −5.30676319704323973598498505434, −3.92116843585832609539625926280, −3.15082815965666084003107400687, −2.31159966609945927097413968976, −1.79533040451211485753299454613, −0.45500974524589911079864599943,
0.45500974524589911079864599943, 1.79533040451211485753299454613, 2.31159966609945927097413968976, 3.15082815965666084003107400687, 3.92116843585832609539625926280, 5.30676319704323973598498505434, 6.12335033972654218210746491807, 6.77795789414029725958832711226, 7.39523340890891233347288630205, 8.044209681276593803743468800061