| L(s) = 1 | + 1.17·3-s + 3.77·5-s − 4.01·7-s − 1.61·9-s + 5.78·11-s + 1.65·13-s + 4.43·15-s − 2.06·17-s + 1.74·19-s − 4.71·21-s − 0.868·23-s + 9.25·25-s − 5.42·27-s + 9.10·29-s + 5.69·31-s + 6.80·33-s − 15.1·35-s + 6.09·37-s + 1.94·39-s + 10.9·41-s − 7.21·43-s − 6.11·45-s − 2.22·47-s + 9.08·49-s − 2.43·51-s − 11.0·53-s + 21.8·55-s + ⋯ |
| L(s) = 1 | + 0.678·3-s + 1.68·5-s − 1.51·7-s − 0.539·9-s + 1.74·11-s + 0.460·13-s + 1.14·15-s − 0.501·17-s + 0.399·19-s − 1.02·21-s − 0.181·23-s + 1.85·25-s − 1.04·27-s + 1.69·29-s + 1.02·31-s + 1.18·33-s − 2.55·35-s + 1.00·37-s + 0.312·39-s + 1.71·41-s − 1.09·43-s − 0.910·45-s − 0.324·47-s + 1.29·49-s − 0.340·51-s − 1.51·53-s + 2.94·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.594709783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.594709783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 3 | \( 1 - 1.17T + 3T^{2} \) |
| 5 | \( 1 - 3.77T + 5T^{2} \) |
| 7 | \( 1 + 4.01T + 7T^{2} \) |
| 11 | \( 1 - 5.78T + 11T^{2} \) |
| 13 | \( 1 - 1.65T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 + 0.868T + 23T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 - 5.69T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 2.22T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 - 0.0497T + 71T^{2} \) |
| 73 | \( 1 + 3.64T + 73T^{2} \) |
| 79 | \( 1 + 5.87T + 79T^{2} \) |
| 83 | \( 1 - 1.98T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 5.72T + 97T^{2} \) |
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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
−9.579315538277604871322398002559, −9.065173035561948827222286356326, −8.317425556405690988698251325450, −6.77156867025326198252961068500, −6.33315579514613955032540821462, −5.82858271987639948952549298522, −4.37451835476606302562839290007, −3.21121335270809836208753006624, −2.56811936323561939009742809961, −1.26351967056821844930212407490,
1.26351967056821844930212407490, 2.56811936323561939009742809961, 3.21121335270809836208753006624, 4.37451835476606302562839290007, 5.82858271987639948952549298522, 6.33315579514613955032540821462, 6.77156867025326198252961068500, 8.317425556405690988698251325450, 9.065173035561948827222286356326, 9.579315538277604871322398002559
