L(s) = 1 | − 2-s − 0.339·3-s − 4-s + 2.88·5-s + 0.339·6-s + 3.54·7-s + 3·8-s − 2.88·9-s − 2.88·10-s − 11-s + 0.339·12-s − 3.54·14-s − 0.980·15-s − 16-s − 5.22·17-s + 2.88·18-s − 6.22·19-s − 2.88·20-s − 1.20·21-s + 22-s + 0.679·23-s − 1.01·24-s + 3.32·25-s + 2·27-s − 3.54·28-s − 8.42·29-s + 0.980·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.196·3-s − 0.5·4-s + 1.28·5-s + 0.138·6-s + 1.33·7-s + 1.06·8-s − 0.961·9-s − 0.912·10-s − 0.301·11-s + 0.0981·12-s − 0.947·14-s − 0.253·15-s − 0.250·16-s − 1.26·17-s + 0.679·18-s − 1.42·19-s − 0.644·20-s − 0.262·21-s + 0.213·22-s + 0.141·23-s − 0.208·24-s + 0.664·25-s + 0.384·27-s − 0.669·28-s − 1.56·29-s + 0.178·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 + 0.339T + 3T^{2} \) |
| 5 | \( 1 - 2.88T + 5T^{2} \) |
| 7 | \( 1 - 3.54T + 7T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 - 0.679T + 23T^{2} \) |
| 29 | \( 1 + 8.42T + 29T^{2} \) |
| 31 | \( 1 + 8.40T + 31T^{2} \) |
| 37 | \( 1 + 3.52T + 37T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 + 6.10T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 4.01T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 0.904T + 79T^{2} \) |
| 83 | \( 1 + 4.90T + 83T^{2} \) |
| 89 | \( 1 - 5.65T + 89T^{2} \) |
| 97 | \( 1 + 2.79T + 97T^{2} \) |
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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
−8.869771500895368738722507669687, −8.347228448982201812833033178597, −7.48756309134970383646291009604, −6.40973160153059417578672052315, −5.48133346358482224854470332919, −4.99600428314235092104159210905, −3.99411177855531016231784473950, −2.24999373050341079973267211389, −1.71059690174772208215974798855, 0,
1.71059690174772208215974798855, 2.24999373050341079973267211389, 3.99411177855531016231784473950, 4.99600428314235092104159210905, 5.48133346358482224854470332919, 6.40973160153059417578672052315, 7.48756309134970383646291009604, 8.347228448982201812833033178597, 8.869771500895368738722507669687