L(s) = 1 | − 0.208·2-s − 0.345·3-s − 1.95·4-s − 1.39·5-s + 0.0720·6-s − 7-s + 0.824·8-s − 2.88·9-s + 0.290·10-s − 2.68·11-s + 0.676·12-s − 5.39·13-s + 0.208·14-s + 0.481·15-s + 3.74·16-s − 1.45·17-s + 0.600·18-s − 6.24·19-s + 2.72·20-s + 0.345·21-s + 0.560·22-s + 2.49·23-s − 0.285·24-s − 3.05·25-s + 1.12·26-s + 2.03·27-s + 1.95·28-s + ⋯ |
L(s) = 1 | − 0.147·2-s − 0.199·3-s − 0.978·4-s − 0.623·5-s + 0.0294·6-s − 0.377·7-s + 0.291·8-s − 0.960·9-s + 0.0918·10-s − 0.810·11-s + 0.195·12-s − 1.49·13-s + 0.0557·14-s + 0.124·15-s + 0.935·16-s − 0.352·17-s + 0.141·18-s − 1.43·19-s + 0.609·20-s + 0.0754·21-s + 0.119·22-s + 0.519·23-s − 0.0581·24-s − 0.611·25-s + 0.220·26-s + 0.391·27-s + 0.369·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.01908833386\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01908833386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
good | 2 | \( 1 + 0.208T + 2T^{2} \) |
| 3 | \( 1 + 0.345T + 3T^{2} \) |
| 5 | \( 1 + 1.39T + 5T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 + 1.45T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 - 2.49T + 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 1.33T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 + 8.65T + 47T^{2} \) |
| 53 | \( 1 - 6.04T + 53T^{2} \) |
| 59 | \( 1 - 4.73T + 59T^{2} \) |
| 61 | \( 1 + 2.18T + 61T^{2} \) |
| 67 | \( 1 - 8.28T + 67T^{2} \) |
| 71 | \( 1 + 7.08T + 71T^{2} \) |
| 73 | \( 1 - 0.930T + 73T^{2} \) |
| 79 | \( 1 + 7.53T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 - 1.07T + 97T^{2} \) |
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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
−8.622115478545796577479340006684, −8.138800790481985135860841584722, −7.36097942488262358107279478141, −6.49869687646201088980343753441, −5.39408388229930330179987600471, −4.97887146652048718171616409823, −4.05171581145997904184680864823, −3.17288191714894554177294695540, −2.13770947916352132090623907844, −0.081022632135031732084269063635,
0.081022632135031732084269063635, 2.13770947916352132090623907844, 3.17288191714894554177294695540, 4.05171581145997904184680864823, 4.97887146652048718171616409823, 5.39408388229930330179987600471, 6.49869687646201088980343753441, 7.36097942488262358107279478141, 8.138800790481985135860841584722, 8.622115478545796577479340006684