| L(s) = 1 | − 7-s − 4·11-s − 6·13-s + 2·17-s + 4·19-s − 8·23-s + 2·29-s + 10·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s + 4·59-s + 6·61-s + 4·67-s + 8·71-s − 10·73-s + 4·77-s + 4·83-s + 6·89-s + 6·91-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 0.371·29-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 0.768·61-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s + 0.439·83-s + 0.635·89-s + 0.628·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.70659764964587, −15.10137009860046, −14.41530418705566, −14.22044460670655, −13.35978730707895, −12.97735713181105, −12.41054791449810, −11.82065827190064, −11.52926327964396, −10.41947962907158, −10.26476601082378, −9.624567245155726, −9.290878361187269, −8.190948290804764, −7.865032727282517, −7.399677190972346, −6.725433637565727, −5.903399551804274, −5.433020615831697, −4.832482907751441, −4.149036223814881, −3.326024620823437, −2.540560804006331, −2.220381897195115, −0.8879255454605991, 0,
0.8879255454605991, 2.220381897195115, 2.540560804006331, 3.326024620823437, 4.149036223814881, 4.832482907751441, 5.433020615831697, 5.903399551804274, 6.725433637565727, 7.399677190972346, 7.865032727282517, 8.190948290804764, 9.290878361187269, 9.624567245155726, 10.26476601082378, 10.41947962907158, 11.52926327964396, 11.82065827190064, 12.41054791449810, 12.97735713181105, 13.35978730707895, 14.22044460670655, 14.41530418705566, 15.10137009860046, 15.70659764964587
