L(s) = 1 | + 3.61·5-s − 3.03·7-s − 1.68·11-s − 4.92·13-s + 1.55·17-s − 5.19·19-s − 2.10·23-s + 8.06·25-s − 2.89·29-s − 3.08·31-s − 10.9·35-s − 0.374·37-s − 11.7·41-s − 3.97·43-s − 6.21·47-s + 2.19·49-s + 6.03·53-s − 6.09·55-s + 59-s + 4.67·61-s − 17.8·65-s + 5.13·67-s − 2.17·71-s − 0.761·73-s + 5.11·77-s + 6.64·79-s + 3.43·83-s + ⋯ |
L(s) = 1 | + 1.61·5-s − 1.14·7-s − 0.508·11-s − 1.36·13-s + 0.376·17-s − 1.19·19-s − 0.438·23-s + 1.61·25-s − 0.537·29-s − 0.554·31-s − 1.85·35-s − 0.0615·37-s − 1.84·41-s − 0.605·43-s − 0.905·47-s + 0.313·49-s + 0.829·53-s − 0.821·55-s + 0.130·59-s + 0.598·61-s − 2.20·65-s + 0.627·67-s − 0.258·71-s − 0.0891·73-s + 0.582·77-s + 0.747·79-s + 0.377·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - T \) |
good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 3.03T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 - 1.55T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 + 2.89T + 29T^{2} \) |
| 31 | \( 1 + 3.08T + 31T^{2} \) |
| 37 | \( 1 + 0.374T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 + 2.17T + 71T^{2} \) |
| 73 | \( 1 + 0.761T + 73T^{2} \) |
| 79 | \( 1 - 6.64T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 - 7.64T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−8.958223582368887095288445909304, −7.942062784690073020936996066821, −6.85895372018921276740647163243, −6.41748522817875476046516584276, −5.52021339289653259207869496189, −4.94351556709050249145604504352, −3.57458952946007664730918543504, −2.55582689815305349084077657710, −1.88278751533071794243890005165, 0,
1.88278751533071794243890005165, 2.55582689815305349084077657710, 3.57458952946007664730918543504, 4.94351556709050249145604504352, 5.52021339289653259207869496189, 6.41748522817875476046516584276, 6.85895372018921276740647163243, 7.942062784690073020936996066821, 8.958223582368887095288445909304