L(s) = 1 | + 2-s + 4-s + 1.41·5-s + 8-s − 3·9-s + 1.41·10-s − 2·11-s − 7.07·13-s + 16-s + 1.41·17-s − 3·18-s + 2.82·19-s + 1.41·20-s − 2·22-s − 6·23-s − 2.99·25-s − 7.07·26-s − 29-s − 1.41·31-s + 32-s + 1.41·34-s − 3·36-s − 4·37-s + 2.82·38-s + 1.41·40-s + 7.07·41-s − 4·43-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.632·5-s + 0.353·8-s − 9-s + 0.447·10-s − 0.603·11-s − 1.96·13-s + 0.250·16-s + 0.342·17-s − 0.707·18-s + 0.648·19-s + 0.316·20-s − 0.426·22-s − 1.25·23-s − 0.599·25-s − 1.38·26-s − 0.185·29-s − 0.254·31-s + 0.176·32-s + 0.242·34-s − 0.5·36-s − 0.657·37-s + 0.458·38-s + 0.223·40-s + 1.10·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.134192960944930627699696299254, −7.69206300215806924993208924290, −6.77477756733577158503456457859, −5.85590396186397951401981424018, −5.34721497262056777570880583713, −4.69835287911109903932785863065, −3.48091181158175284480969721410, −2.62397009510143994678621541863, −1.94548932682159096552640825297, 0,
1.94548932682159096552640825297, 2.62397009510143994678621541863, 3.48091181158175284480969721410, 4.69835287911109903932785863065, 5.34721497262056777570880583713, 5.85590396186397951401981424018, 6.77477756733577158503456457859, 7.69206300215806924993208924290, 8.134192960944930627699696299254