L(s) = 1 | − 2-s + 2.82·3-s + 4-s − 5-s − 2.82·6-s + 7-s − 8-s + 5.00·9-s + 10-s + 2.82·12-s − 2·13-s − 14-s − 2.82·15-s + 16-s − 7.65·17-s − 5.00·18-s + 2.82·19-s − 20-s + 2.82·21-s − 2.82·23-s − 2.82·24-s + 25-s + 2·26-s + 5.65·27-s + 28-s − 3.17·29-s + 2.82·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.63·3-s + 0.5·4-s − 0.447·5-s − 1.15·6-s + 0.377·7-s − 0.353·8-s + 1.66·9-s + 0.316·10-s + 0.816·12-s − 0.554·13-s − 0.267·14-s − 0.730·15-s + 0.250·16-s − 1.85·17-s − 1.17·18-s + 0.648·19-s − 0.223·20-s + 0.617·21-s − 0.589·23-s − 0.577·24-s + 0.200·25-s + 0.392·26-s + 1.08·27-s + 0.188·28-s − 0.588·29-s + 0.516·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 9.31T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 - 5.17T + 79T^{2} \) |
| 83 | \( 1 - 2.34T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 0.828T + 97T^{2} \) |
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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
−7.69290706061046991419729170522, −7.12410196158094950650297801529, −6.41715937623756269132467792109, −5.27176126435502999331733538728, −4.33114404947085658146874566075, −3.78682559478378396781756919967, −2.82989658276470805211087713559, −2.25843032292322031240915289699, −1.50404889698204852610377958237, 0,
1.50404889698204852610377958237, 2.25843032292322031240915289699, 2.82989658276470805211087713559, 3.78682559478378396781756919967, 4.33114404947085658146874566075, 5.27176126435502999331733538728, 6.41715937623756269132467792109, 7.12410196158094950650297801529, 7.69290706061046991419729170522