L(s) = 1 | − 3.12·3-s + 6.76·9-s + 2.48·11-s + 4.15·13-s − 5.76·17-s + 1.60·19-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 + 18.0·51-s − 9.21·53-s − 5.03·57-s + 5.92·59-s − 5.35·61-s − 7.52·67-s − 22.7·69-s − 4.24·71-s − 7.28·73-s + 16.9·79-s + 16.4·81-s + ⋯ |
L(s) = 1 | − 1.80·3-s + 2.25·9-s + 0.749·11-s + 1.15·13-s − 1.39·17-s + 0.369·19-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 + 2.52·51-s − 1.26·53-s − 0.666·57-s + 0.770·59-s − 0.686·61-s − 0.919·67-s − 2.73·69-s − 0.504·71-s − 0.852·73-s + 1.91·79-s + 1.82·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
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
−6.83060634260935490854640187992, −6.63086223981356413318771171877, −6.14602813777282541664424761077, −5.12498263918387015289198724300, −4.88061085036576032263737559258, −3.99542494216663733990644151862, −3.21976239501058502223547587228, −1.73670139120075347509436197335, −1.09446416799895602715214706682, 0,
1.09446416799895602715214706682, 1.73670139120075347509436197335, 3.21976239501058502223547587228, 3.99542494216663733990644151862, 4.88061085036576032263737559258, 5.12498263918387015289198724300, 6.14602813777282541664424761077, 6.63086223981356413318771171877, 6.83060634260935490854640187992