L(s) = 1 | − 2-s − 3.00·3-s + 4-s − 2.58·5-s + 3.00·6-s − 5.14·7-s − 8-s + 6.00·9-s + 2.58·10-s + 0.908·11-s − 3.00·12-s − 2.31·13-s + 5.14·14-s + 7.75·15-s + 16-s − 0.446·17-s − 6.00·18-s + 19-s − 2.58·20-s + 15.4·21-s − 0.908·22-s − 6.32·23-s + 3.00·24-s + 1.68·25-s + 2.31·26-s − 9.01·27-s − 5.14·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 0.5·4-s − 1.15·5-s + 1.22·6-s − 1.94·7-s − 0.353·8-s + 2.00·9-s + 0.817·10-s + 0.273·11-s − 0.866·12-s − 0.642·13-s + 1.37·14-s + 2.00·15-s + 0.250·16-s − 0.108·17-s − 1.41·18-s + 0.229·19-s − 0.578·20-s + 3.36·21-s − 0.193·22-s − 1.31·23-s + 0.612·24-s + 0.336·25-s + 0.454·26-s − 1.73·27-s − 0.971·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 + 3.00T + 3T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 7 | \( 1 + 5.14T + 7T^{2} \) |
| 11 | \( 1 - 0.908T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 + 0.446T + 17T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 + 0.259T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 - 1.48T + 53T^{2} \) |
| 59 | \( 1 + 5.22T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 - 4.83T + 67T^{2} \) |
| 71 | \( 1 + 0.367T + 71T^{2} \) |
| 73 | \( 1 - 5.00T + 73T^{2} \) |
| 79 | \( 1 + 5.38T + 79T^{2} \) |
| 83 | \( 1 - 5.97T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−7.26183513190822550684002376026, −6.83678849499233884285304390928, −6.21401755917826110110683089805, −5.67182014290581730065708374630, −4.75370887161271968537880777492, −3.80272813860024867660530920589, −3.35083677308552905702281545355, −1.95489348704238985140081238539, −0.53190022893470863807229356927, 0,
0.53190022893470863807229356927, 1.95489348704238985140081238539, 3.35083677308552905702281545355, 3.80272813860024867660530920589, 4.75370887161271968537880777492, 5.67182014290581730065708374630, 6.21401755917826110110683089805, 6.83678849499233884285304390928, 7.26183513190822550684002376026