L(s) = 1 | + 2-s − 0.369·3-s + 4-s + 5-s − 0.369·6-s + 4.16·7-s + 8-s − 2.86·9-s + 10-s − 5.82·11-s − 0.369·12-s − 5.41·13-s + 4.16·14-s − 0.369·15-s + 16-s − 3.34·17-s − 2.86·18-s + 1.75·19-s + 20-s − 1.53·21-s − 5.82·22-s + 1.42·23-s − 0.369·24-s + 25-s − 5.41·26-s + 2.16·27-s + 4.16·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.213·3-s + 0.5·4-s + 0.447·5-s − 0.150·6-s + 1.57·7-s + 0.353·8-s − 0.954·9-s + 0.316·10-s − 1.75·11-s − 0.106·12-s − 1.50·13-s + 1.11·14-s − 0.0953·15-s + 0.250·16-s − 0.811·17-s − 0.674·18-s + 0.402·19-s + 0.223·20-s − 0.335·21-s − 1.24·22-s + 0.297·23-s − 0.0753·24-s + 0.200·25-s − 1.06·26-s + 0.416·27-s + 0.786·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 0.369T + 3T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 17 | \( 1 + 3.34T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 - 0.619T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 + 0.960T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 + 2.02T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 9.76T + 61T^{2} \) |
| 67 | \( 1 - 7.72T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 + 7.65T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 9.11T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−7.80110179394539371055740142702, −7.03299424885819491149305958850, −6.05599104463914251060979121363, −5.32792793850577420304013572448, −4.91607300012698081941212197560, −4.49981748915334164583039269030, −2.85424768843153241687971640864, −2.59780507756580884590120047144, −1.61391756827453644507911255741, 0,
1.61391756827453644507911255741, 2.59780507756580884590120047144, 2.85424768843153241687971640864, 4.49981748915334164583039269030, 4.91607300012698081941212197560, 5.32792793850577420304013572448, 6.05599104463914251060979121363, 7.03299424885819491149305958850, 7.80110179394539371055740142702