L(s) = 1 | − 0.959·2-s − 1.07·4-s + 2.76·5-s + 2.95·7-s + 2.95·8-s − 2.64·10-s − 1.28·11-s + 5.35·13-s − 2.83·14-s − 0.675·16-s + 5.11·17-s − 7.15·19-s − 2.98·20-s + 1.22·22-s + 23-s + 2.62·25-s − 5.14·26-s − 3.18·28-s − 29-s + 6.82·31-s − 5.26·32-s − 4.91·34-s + 8.15·35-s + 1.42·37-s + 6.86·38-s + 8.15·40-s + 0.739·41-s + ⋯ |
L(s) = 1 | − 0.678·2-s − 0.539·4-s + 1.23·5-s + 1.11·7-s + 1.04·8-s − 0.837·10-s − 0.386·11-s + 1.48·13-s − 0.757·14-s − 0.168·16-s + 1.24·17-s − 1.64·19-s − 0.666·20-s + 0.262·22-s + 0.208·23-s + 0.524·25-s − 1.00·26-s − 0.602·28-s − 0.185·29-s + 1.22·31-s − 0.930·32-s − 0.842·34-s + 1.37·35-s + 0.234·37-s + 1.11·38-s + 1.28·40-s + 0.115·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.026825952\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.026825952\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 0.959T + 2T^{2} \) |
| 5 | \( 1 - 2.76T + 5T^{2} \) |
| 7 | \( 1 - 2.95T + 7T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 5.11T + 17T^{2} \) |
| 19 | \( 1 + 7.15T + 19T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 - 0.739T + 41T^{2} \) |
| 43 | \( 1 - 8.76T + 43T^{2} \) |
| 47 | \( 1 + 4.04T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 1.57T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 8.17T + 67T^{2} \) |
| 71 | \( 1 - 8.05T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−8.128143957707358313614604184657, −7.78054706754248768317355888919, −6.59391066793036995852177150657, −5.90071260591055631983346114295, −5.29503047766792720935361334081, −4.52085017972384659597064668508, −3.76310134304346775057357190443, −2.45279879858140484219492290229, −1.60177176416839697905617498584, −0.931804509630557166833825478700,
0.931804509630557166833825478700, 1.60177176416839697905617498584, 2.45279879858140484219492290229, 3.76310134304346775057357190443, 4.52085017972384659597064668508, 5.29503047766792720935361334081, 5.90071260591055631983346114295, 6.59391066793036995852177150657, 7.78054706754248768317355888919, 8.128143957707358313614604184657