| L(s) = 1 | − 1.21·2-s − 1.27·3-s − 0.512·4-s − 4.16·5-s + 1.55·6-s − 2.00·7-s + 3.06·8-s − 1.38·9-s + 5.07·10-s + 1.67·11-s + 0.651·12-s − 3.62·13-s + 2.44·14-s + 5.29·15-s − 2.71·16-s − 2.04·17-s + 1.68·18-s + 7.36·19-s + 2.13·20-s + 2.54·21-s − 2.04·22-s − 3.41·23-s − 3.89·24-s + 12.3·25-s + 4.41·26-s + 5.57·27-s + 1.02·28-s + ⋯ |
| L(s) = 1 | − 0.862·2-s − 0.734·3-s − 0.256·4-s − 1.86·5-s + 0.633·6-s − 0.756·7-s + 1.08·8-s − 0.461·9-s + 1.60·10-s + 0.504·11-s + 0.187·12-s − 1.00·13-s + 0.652·14-s + 1.36·15-s − 0.678·16-s − 0.494·17-s + 0.397·18-s + 1.68·19-s + 0.476·20-s + 0.555·21-s − 0.435·22-s − 0.712·23-s − 0.795·24-s + 2.46·25-s + 0.866·26-s + 1.07·27-s + 0.193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01638223808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01638223808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 8011 | \( 1+O(T) \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 + 3.36T + 31T^{2} \) |
| 37 | \( 1 - 0.144T + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 - 9.04T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 1.59T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.32T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.31T + 97T^{2} \) |
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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
−7.71477226884824306828302445484, −7.33079971007243580362367341664, −6.80510078797537051168289922377, −5.69861421800774763944426107350, −5.00085186526839180215653896362, −4.25173608317857367393223078009, −3.62491188767385624968480004441, −2.78997571953737197275975041394, −1.20455087472108570311306221793, −0.086631000953206225179095352457,
0.086631000953206225179095352457, 1.20455087472108570311306221793, 2.78997571953737197275975041394, 3.62491188767385624968480004441, 4.25173608317857367393223078009, 5.00085186526839180215653896362, 5.69861421800774763944426107350, 6.80510078797537051168289922377, 7.33079971007243580362367341664, 7.71477226884824306828302445484
