L(s) = 1 | − 1.71·2-s + 0.936·4-s + 4.13·5-s + 0.967·7-s + 1.82·8-s − 7.08·10-s + 6.54·11-s + 2.28·13-s − 1.65·14-s − 4.99·16-s + 2.37·17-s + 2.94·19-s + 3.87·20-s − 11.2·22-s + 1.52·23-s + 12.0·25-s − 3.91·26-s + 0.906·28-s − 2.59·29-s − 4.67·31-s + 4.91·32-s − 4.07·34-s + 4·35-s − 9.81·37-s − 5.03·38-s + 7.53·40-s − 4.37·41-s + ⋯ |
L(s) = 1 | − 1.21·2-s + 0.468·4-s + 1.84·5-s + 0.365·7-s + 0.644·8-s − 2.24·10-s + 1.97·11-s + 0.632·13-s − 0.443·14-s − 1.24·16-s + 0.576·17-s + 0.674·19-s + 0.866·20-s − 2.39·22-s + 0.317·23-s + 2.41·25-s − 0.767·26-s + 0.171·28-s − 0.481·29-s − 0.840·31-s + 0.869·32-s − 0.698·34-s + 0.676·35-s − 1.61·37-s − 0.817·38-s + 1.19·40-s − 0.682·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.468586721\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.468586721\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 7 | \( 1 - 0.967T + 7T^{2} \) |
| 11 | \( 1 - 6.54T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 17 | \( 1 - 2.37T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 - 1.52T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 + 4.67T + 31T^{2} \) |
| 37 | \( 1 + 9.81T + 37T^{2} \) |
| 41 | \( 1 + 4.37T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 3.66T + 59T^{2} \) |
| 61 | \( 1 + 0.791T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 6.43T + 71T^{2} \) |
| 73 | \( 1 + 3.78T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 3.28T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−9.688071566202146880429660846752, −9.007568843498402267738629403823, −8.672796452244273995463424978001, −7.34007248964104204904397808289, −6.58733657214333798063424164442, −5.77790215072254009820905041215, −4.79496503759200843834659750692, −3.41031719193765221419338716851, −1.65840649958548290235169214077, −1.40851084851587041099492579796,
1.40851084851587041099492579796, 1.65840649958548290235169214077, 3.41031719193765221419338716851, 4.79496503759200843834659750692, 5.77790215072254009820905041215, 6.58733657214333798063424164442, 7.34007248964104204904397808289, 8.672796452244273995463424978001, 9.007568843498402267738629403823, 9.688071566202146880429660846752