| L(s) = 1 | + 2.18·2-s + 2.75·4-s + 3.64·5-s + 2.94·7-s + 1.64·8-s + 7.95·10-s + 2.90·11-s + 1.83·13-s + 6.42·14-s − 1.91·16-s − 6.16·17-s − 7.26·19-s + 10.0·20-s + 6.32·22-s + 23-s + 8.30·25-s + 4.00·26-s + 8.12·28-s + 29-s + 9.78·31-s − 7.47·32-s − 13.4·34-s + 10.7·35-s − 1.23·37-s − 15.8·38-s + 6.01·40-s + 6.05·41-s + ⋯ |
| L(s) = 1 | + 1.54·2-s + 1.37·4-s + 1.63·5-s + 1.11·7-s + 0.582·8-s + 2.51·10-s + 0.874·11-s + 0.508·13-s + 1.71·14-s − 0.479·16-s − 1.49·17-s − 1.66·19-s + 2.24·20-s + 1.34·22-s + 0.208·23-s + 1.66·25-s + 0.784·26-s + 1.53·28-s + 0.185·29-s + 1.75·31-s − 1.32·32-s − 2.30·34-s + 1.81·35-s − 0.203·37-s − 2.56·38-s + 0.950·40-s + 0.946·41-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
| \(L(1)\) |
\(\approx\) |
\(7.801347065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.801347065\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.18T + 2T^{2} \) |
| 5 | \( 1 - 3.64T + 5T^{2} \) |
| 7 | \( 1 - 2.94T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 19 | \( 1 + 7.26T + 19T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 - 3.54T + 59T^{2} \) |
| 61 | \( 1 + 9.97T + 61T^{2} \) |
| 67 | \( 1 - 5.17T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 3.11T + 89T^{2} \) |
| 97 | \( 1 + 1.36T + 97T^{2} \) |
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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
−8.093229510736624522484357342498, −6.76549926867860961559462290082, −6.43464119334507956244430392327, −5.93821601034343602507068916069, −5.11536543732855460966125831791, −4.45332864383794436876666149293, −4.03809650224928353583015550884, −2.61861236548015976434403674574, −2.20471086479237356671227308864, −1.32449471375283707383060256271,
1.32449471375283707383060256271, 2.20471086479237356671227308864, 2.61861236548015976434403674574, 4.03809650224928353583015550884, 4.45332864383794436876666149293, 5.11536543732855460966125831791, 5.93821601034343602507068916069, 6.43464119334507956244430392327, 6.76549926867860961559462290082, 8.093229510736624522484357342498
