| L(s) = 1 | + 3.12·3-s − 7-s + 6.76·9-s + 2.48·11-s − 4.15·13-s + 5.76·17-s − 1.60·19-s − 3.12·21-s + 7.28·23-s + 11.7·27-s + 1.45·29-s − 2.24·31-s + 7.76·33-s − 6·37-s − 12.9·39-s + 11.2·41-s − 5.28·43-s + 3.45·47-s + 49-s + 18.0·51-s − 9.21·53-s − 5.03·57-s − 5.92·59-s + 5.35·61-s − 6.76·63-s − 7.52·67-s + 22.7·69-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 0.377·7-s + 2.25·9-s + 0.749·11-s − 1.15·13-s + 1.39·17-s − 0.369·19-s − 0.681·21-s + 1.51·23-s + 2.26·27-s + 0.270·29-s − 0.404·31-s + 1.35·33-s − 0.986·37-s − 2.07·39-s + 1.76·41-s − 0.805·43-s + 0.503·47-s + 0.142·49-s + 2.52·51-s − 1.26·53-s − 0.666·57-s − 0.770·59-s + 0.686·61-s − 0.852·63-s − 0.919·67-s + 2.73·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.239012867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.239012867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 11 | \( 1 - 2.48T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 7.28T + 23T^{2} \) |
| 29 | \( 1 - 1.45T + 29T^{2} \) |
| 31 | \( 1 + 2.24T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 + 5.92T + 59T^{2} \) |
| 61 | \( 1 - 5.35T + 61T^{2} \) |
| 67 | \( 1 + 7.52T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 - 7.28T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
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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
−9.409344069834454129631226393445, −8.901213072744211247888683387755, −7.956467123206446196136747544306, −7.35235238807244672391641168005, −6.62310445893279308196615686653, −5.22010502321518347905447888820, −4.16965028885278454720126471453, −3.28623503642047064414362320420, −2.62002873284730933572755918240, −1.38865274016735659785868412142,
1.38865274016735659785868412142, 2.62002873284730933572755918240, 3.28623503642047064414362320420, 4.16965028885278454720126471453, 5.22010502321518347905447888820, 6.62310445893279308196615686653, 7.35235238807244672391641168005, 7.956467123206446196136747544306, 8.901213072744211247888683387755, 9.409344069834454129631226393445
