| L(s) = 1 | + 2.17·3-s − 4.91·7-s + 1.74·9-s − 2.91·11-s + 3.09·13-s + 17-s + 5.61·19-s − 10.7·21-s + 0.918·23-s − 2.74·27-s + 7.61·29-s + 8.17·31-s − 6.35·33-s + 6.35·37-s + 6.74·39-s + 3.83·41-s − 11.8·43-s − 4.74·47-s + 17.1·49-s + 2.17·51-s + 7.25·53-s + 12.2·57-s + 1.61·59-s − 0.741·61-s − 8.56·63-s − 6.70·67-s + 1.99·69-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 1.85·7-s + 0.580·9-s − 0.879·11-s + 0.858·13-s + 0.242·17-s + 1.28·19-s − 2.33·21-s + 0.191·23-s − 0.527·27-s + 1.41·29-s + 1.46·31-s − 1.10·33-s + 1.04·37-s + 1.07·39-s + 0.599·41-s − 1.80·43-s − 0.691·47-s + 2.45·49-s + 0.304·51-s + 0.997·53-s + 1.61·57-s + 0.210·59-s − 0.0948·61-s − 1.07·63-s − 0.819·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.315368663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.315368663\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 19 | \( 1 - 5.61T + 19T^{2} \) |
| 23 | \( 1 - 0.918T + 23T^{2} \) |
| 29 | \( 1 - 7.61T + 29T^{2} \) |
| 31 | \( 1 - 8.17T + 31T^{2} \) |
| 37 | \( 1 - 6.35T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 - 7.25T + 53T^{2} \) |
| 59 | \( 1 - 1.61T + 59T^{2} \) |
| 61 | \( 1 + 0.741T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 5.09T + 73T^{2} \) |
| 79 | \( 1 + 7.62T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 4.22T + 89T^{2} \) |
| 97 | \( 1 - 2.22T + 97T^{2} \) |
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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
−8.580748888480470387741130672808, −8.024738043316794479856553328056, −7.20782725744183354243360178054, −6.41526876605023906915039182516, −5.76310881799783581892089486255, −4.62593510531740992783941516767, −3.42670000767634148083004838509, −3.15995877619237638829712456415, −2.43942481041199752944537355615, −0.838703035409364349315896699644,
0.838703035409364349315896699644, 2.43942481041199752944537355615, 3.15995877619237638829712456415, 3.42670000767634148083004838509, 4.62593510531740992783941516767, 5.76310881799783581892089486255, 6.41526876605023906915039182516, 7.20782725744183354243360178054, 8.024738043316794479856553328056, 8.580748888480470387741130672808
