L(s) = 1 | − 3.38·3-s − 2.49·5-s − 4.68·7-s + 8.42·9-s + 11-s − 0.240·13-s + 8.41·15-s − 3.99·17-s − 0.972·19-s + 15.8·21-s − 8.41·23-s + 1.20·25-s − 18.3·27-s − 5.63·29-s + 8.04·31-s − 3.38·33-s + 11.6·35-s − 1.83·37-s + 0.813·39-s − 3.11·41-s + 9.84·43-s − 20.9·45-s − 4.63·47-s + 14.9·49-s + 13.5·51-s + 10.2·53-s − 2.49·55-s + ⋯ |
L(s) = 1 | − 1.95·3-s − 1.11·5-s − 1.77·7-s + 2.80·9-s + 0.301·11-s − 0.0667·13-s + 2.17·15-s − 0.969·17-s − 0.223·19-s + 3.45·21-s − 1.75·23-s + 0.240·25-s − 3.52·27-s − 1.04·29-s + 1.44·31-s − 0.588·33-s + 1.97·35-s − 0.301·37-s + 0.130·39-s − 0.487·41-s + 1.50·43-s − 3.12·45-s − 0.675·47-s + 2.14·49-s + 1.89·51-s + 1.40·53-s − 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
good | 3 | \( 1 + 3.38T + 3T^{2} \) |
| 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 13 | \( 1 + 0.240T + 13T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 19 | \( 1 + 0.972T + 19T^{2} \) |
| 23 | \( 1 + 8.41T + 23T^{2} \) |
| 29 | \( 1 + 5.63T + 29T^{2} \) |
| 31 | \( 1 - 8.04T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 + 3.11T + 41T^{2} \) |
| 43 | \( 1 - 9.84T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 11.0T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 + 1.15T + 89T^{2} \) |
| 97 | \( 1 + 7.94T + 97T^{2} \) |
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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
−7.38970434160367087577183767380, −6.81789138795369043967304538309, −6.27345514988472830981649793409, −5.83504100095140428792218130230, −4.83412020625721611512388982545, −3.97159976411004867126716082198, −3.74703475537709715248228411063, −2.21591211123507244217742047931, −0.67147552781478844562074909174, 0,
0.67147552781478844562074909174, 2.21591211123507244217742047931, 3.74703475537709715248228411063, 3.97159976411004867126716082198, 4.83412020625721611512388982545, 5.83504100095140428792218130230, 6.27345514988472830981649793409, 6.81789138795369043967304538309, 7.38970434160367087577183767380