L(s) = 1 | + 2.26·2-s + 1.82·3-s + 3.13·4-s + 2.21·5-s + 4.13·6-s + 2.57·8-s + 0.329·9-s + 5.01·10-s − 3.49·11-s + 5.72·12-s − 2.53·13-s + 4.03·15-s − 0.441·16-s − 17-s + 0.747·18-s + 6.37·19-s + 6.93·20-s − 7.91·22-s − 5.03·23-s + 4.69·24-s − 0.105·25-s − 5.73·26-s − 4.87·27-s − 1.84·29-s + 9.14·30-s + 9.70·31-s − 6.14·32-s + ⋯ |
L(s) = 1 | + 1.60·2-s + 1.05·3-s + 1.56·4-s + 0.989·5-s + 1.68·6-s + 0.909·8-s + 0.109·9-s + 1.58·10-s − 1.05·11-s + 1.65·12-s − 0.702·13-s + 1.04·15-s − 0.110·16-s − 0.242·17-s + 0.176·18-s + 1.46·19-s + 1.55·20-s − 1.68·22-s − 1.05·23-s + 0.958·24-s − 0.0210·25-s − 1.12·26-s − 0.937·27-s − 0.342·29-s + 1.67·30-s + 1.74·31-s − 1.08·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.289965310\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.289965310\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - 2.26T + 2T^{2} \) |
| 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 2.53T + 13T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 - 4.02T + 37T^{2} \) |
| 41 | \( 1 + 0.0679T + 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 2.87T + 59T^{2} \) |
| 61 | \( 1 - 0.0679T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 2.66T + 83T^{2} \) |
| 89 | \( 1 + 4.22T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−9.996320712679190898496053763696, −9.636179276591721155158886097740, −8.359672704683623283729114379176, −7.57039897089906784253872678853, −6.45805398968532023353748388728, −5.55196115099217805270248099062, −4.94124148563921339853556425718, −3.69862756456210015081397094753, −2.72441634463410509054113243579, −2.18307472422679811969296551652,
2.18307472422679811969296551652, 2.72441634463410509054113243579, 3.69862756456210015081397094753, 4.94124148563921339853556425718, 5.55196115099217805270248099062, 6.45805398968532023353748388728, 7.57039897089906784253872678853, 8.359672704683623283729114379176, 9.636179276591721155158886097740, 9.996320712679190898496053763696