| L(s) = 1 | + 2-s − 3.25·3-s + 4-s − 1.47·5-s − 3.25·6-s + 0.915·7-s + 8-s + 7.58·9-s − 1.47·10-s + 0.598·11-s − 3.25·12-s + 3.86·13-s + 0.915·14-s + 4.80·15-s + 16-s − 3.18·17-s + 7.58·18-s − 3.26·19-s − 1.47·20-s − 2.97·21-s + 0.598·22-s − 4.77·23-s − 3.25·24-s − 2.82·25-s + 3.86·26-s − 14.9·27-s + 0.915·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.87·3-s + 0.5·4-s − 0.660·5-s − 1.32·6-s + 0.346·7-s + 0.353·8-s + 2.52·9-s − 0.466·10-s + 0.180·11-s − 0.939·12-s + 1.07·13-s + 0.244·14-s + 1.24·15-s + 0.250·16-s − 0.771·17-s + 1.78·18-s − 0.749·19-s − 0.330·20-s − 0.650·21-s + 0.127·22-s − 0.996·23-s − 0.664·24-s − 0.564·25-s + 0.757·26-s − 2.87·27-s + 0.173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 0.915T + 7T^{2} \) |
| 11 | \( 1 - 0.598T + 11T^{2} \) |
| 13 | \( 1 - 3.86T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 + 4.16T + 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 4.75T + 41T^{2} \) |
| 43 | \( 1 + 1.78T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 + 7.15T + 53T^{2} \) |
| 59 | \( 1 - 5.73T + 59T^{2} \) |
| 61 | \( 1 + 7.81T + 61T^{2} \) |
| 67 | \( 1 + 0.903T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 0.144T + 73T^{2} \) |
| 79 | \( 1 - 6.66T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 - 6.34T + 97T^{2} \) |
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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
−7.58005225392550429540693716918, −6.53787316651272181715463161639, −6.32208033630772654245902631685, −5.62317561125963078102351640084, −4.83274696438459941440966274303, −4.15517861495314774432612143957, −3.79562276244764129059621419053, −2.17621868871711934291098066734, −1.17020207605287993410805943860, 0,
1.17020207605287993410805943860, 2.17621868871711934291098066734, 3.79562276244764129059621419053, 4.15517861495314774432612143957, 4.83274696438459941440966274303, 5.62317561125963078102351640084, 6.32208033630772654245902631685, 6.53787316651272181715463161639, 7.58005225392550429540693716918
