| 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
| \(L(1)\) |
\(\approx\) |
\(1.707632783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707632783\) |
| \(L(\frac{3}{2})\) |
|
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
−8.159346530320855484499369541716, −7.53185504581339010017999542622, −6.29658191590602691918146615156, −6.13426079215153995207914256223, −4.77674224455923128937578073636, −4.39325941772177481192245701712, −3.99434877009151179247254234566, −3.09697815535962458880776520838, −1.88589889064690968804235050383, −0.62615885409951791051554626427,
0.62615885409951791051554626427, 1.88589889064690968804235050383, 3.09697815535962458880776520838, 3.99434877009151179247254234566, 4.39325941772177481192245701712, 4.77674224455923128937578073636, 6.13426079215153995207914256223, 6.29658191590602691918146615156, 7.53185504581339010017999542622, 8.159346530320855484499369541716
