| L(s) = 1 | + 0.409·2-s − 2.84·3-s − 1.83·4-s + 2.40·5-s − 1.16·6-s − 1.56·8-s + 5.10·9-s + 0.983·10-s − 0.464·11-s + 5.21·12-s − 2.07·13-s − 6.83·15-s + 3.02·16-s + 1.51·17-s + 2.08·18-s − 6.50·19-s − 4.40·20-s − 0.190·22-s + 3.71·23-s + 4.46·24-s + 0.767·25-s − 0.850·26-s − 5.98·27-s + 1.83·29-s − 2.79·30-s + 2.31·31-s + 4.37·32-s + ⋯ |
| L(s) = 1 | + 0.289·2-s − 1.64·3-s − 0.916·4-s + 1.07·5-s − 0.475·6-s − 0.554·8-s + 1.70·9-s + 0.310·10-s − 0.140·11-s + 1.50·12-s − 0.575·13-s − 1.76·15-s + 0.755·16-s + 0.367·17-s + 0.492·18-s − 1.49·19-s − 0.983·20-s − 0.0405·22-s + 0.774·23-s + 0.911·24-s + 0.153·25-s − 0.166·26-s − 1.15·27-s + 0.341·29-s − 0.511·30-s + 0.416·31-s + 0.773·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.409T + 2T^{2} \) |
| 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 - 2.40T + 5T^{2} \) |
| 11 | \( 1 + 0.464T + 11T^{2} \) |
| 13 | \( 1 + 2.07T + 13T^{2} \) |
| 17 | \( 1 - 1.51T + 17T^{2} \) |
| 19 | \( 1 + 6.50T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 - 9.96T + 37T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 9.84T + 47T^{2} \) |
| 53 | \( 1 - 1.60T + 53T^{2} \) |
| 59 | \( 1 + 2.50T + 59T^{2} \) |
| 61 | \( 1 + 0.0952T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 15.9T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 - 7.91T + 79T^{2} \) |
| 83 | \( 1 + 8.30T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 6.46T + 97T^{2} \) |
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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
−8.998846838001913564044569267915, −7.942522775067301428536794778225, −6.83028222087847807471804443275, −6.00698614656509015153709095931, −5.70238784880487698757720921806, −4.74642877096576466171246336354, −4.33642527024137610328358059111, −2.74905300835946266266699731535, −1.29470927201546675427199286336, 0,
1.29470927201546675427199286336, 2.74905300835946266266699731535, 4.33642527024137610328358059111, 4.74642877096576466171246336354, 5.70238784880487698757720921806, 6.00698614656509015153709095931, 6.83028222087847807471804443275, 7.942522775067301428536794778225, 8.998846838001913564044569267915
