L(s) = 1 | + 2-s + 2.67·3-s + 4-s − 0.452·5-s + 2.67·6-s − 2.46·7-s + 8-s + 4.14·9-s − 0.452·10-s − 5.22·11-s + 2.67·12-s − 2.02·13-s − 2.46·14-s − 1.20·15-s + 16-s + 1.39·17-s + 4.14·18-s − 0.407·19-s − 0.452·20-s − 6.59·21-s − 5.22·22-s − 6.04·23-s + 2.67·24-s − 4.79·25-s − 2.02·26-s + 3.05·27-s − 2.46·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.54·3-s + 0.5·4-s − 0.202·5-s + 1.09·6-s − 0.933·7-s + 0.353·8-s + 1.38·9-s − 0.143·10-s − 1.57·11-s + 0.771·12-s − 0.562·13-s − 0.659·14-s − 0.312·15-s + 0.250·16-s + 0.339·17-s + 0.976·18-s − 0.0935·19-s − 0.101·20-s − 1.44·21-s − 1.11·22-s − 1.26·23-s + 0.545·24-s − 0.959·25-s − 0.398·26-s + 0.587·27-s − 0.466·28-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 | \( 1 - 2.67T + 3T^{2} \) |
| 5 | \( 1 + 0.452T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 5.22T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 17 | \( 1 - 1.39T + 17T^{2} \) |
| 19 | \( 1 + 0.407T + 19T^{2} \) |
| 23 | \( 1 + 6.04T + 23T^{2} \) |
| 29 | \( 1 - 0.0675T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 + 0.940T + 37T^{2} \) |
| 41 | \( 1 - 4.51T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 4.89T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 6.79T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 - 4.62T + 83T^{2} \) |
| 89 | \( 1 - 7.53T + 89T^{2} \) |
| 97 | \( 1 - 5.52T + 97T^{2} \) |
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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
−7.73731499329336622414512993847, −7.27916391237143290447091579242, −6.26096142636526555277604018203, −5.55428862942800107746144654632, −4.64302485704181142905015908159, −3.80310422237439194288633319571, −3.21606463120828434239893760202, −2.55897571426714342356227944015, −1.92737704475468732450740716188, 0,
1.92737704475468732450740716188, 2.55897571426714342356227944015, 3.21606463120828434239893760202, 3.80310422237439194288633319571, 4.64302485704181142905015908159, 5.55428862942800107746144654632, 6.26096142636526555277604018203, 7.27916391237143290447091579242, 7.73731499329336622414512993847