| L(s) = 1 | − 2.21·2-s − 3-s + 2.89·4-s + 2.02·5-s + 2.21·6-s − 4.79·7-s − 1.98·8-s + 9-s − 4.48·10-s − 4.23·11-s − 2.89·12-s + 13-s + 10.6·14-s − 2.02·15-s − 1.39·16-s − 0.757·17-s − 2.21·18-s + 0.00580·19-s + 5.87·20-s + 4.79·21-s + 9.38·22-s + 0.802·23-s + 1.98·24-s − 0.884·25-s − 2.21·26-s − 27-s − 13.8·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.44·4-s + 0.907·5-s + 0.903·6-s − 1.81·7-s − 0.703·8-s + 0.333·9-s − 1.41·10-s − 1.27·11-s − 0.836·12-s + 0.277·13-s + 2.83·14-s − 0.523·15-s − 0.349·16-s − 0.183·17-s − 0.521·18-s + 0.00133·19-s + 1.31·20-s + 1.04·21-s + 2.00·22-s + 0.167·23-s + 0.405·24-s − 0.176·25-s − 0.434·26-s − 0.192·27-s − 2.62·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 - 2.02T + 5T^{2} \) |
| 7 | \( 1 + 4.79T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 17 | \( 1 + 0.757T + 17T^{2} \) |
| 19 | \( 1 - 0.00580T + 19T^{2} \) |
| 23 | \( 1 - 0.802T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 - 5.37T + 31T^{2} \) |
| 37 | \( 1 - 5.34T + 37T^{2} \) |
| 41 | \( 1 - 9.84T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 - 5.11T + 47T^{2} \) |
| 53 | \( 1 - 4.25T + 53T^{2} \) |
| 59 | \( 1 + 6.12T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 - 0.171T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 4.94T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 + 1.79T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−8.182181649111357325399225335394, −7.39019121681421710785223300170, −6.66124531213176185053029201139, −6.12700713626885907542600969543, −5.48278683725468866397961783870, −4.27687854385498717882656398743, −2.91014381129496441369994006789, −2.33377053741201876895245683022, −0.981536991568862011593332169221, 0,
0.981536991568862011593332169221, 2.33377053741201876895245683022, 2.91014381129496441369994006789, 4.27687854385498717882656398743, 5.48278683725468866397961783870, 6.12700713626885907542600969543, 6.66124531213176185053029201139, 7.39019121681421710785223300170, 8.182181649111357325399225335394
