| L(s) = 1 | + 2-s + 4-s + (1.85 + 1.25i)5-s + 4.78i·7-s + 8-s + (1.85 + 1.25i)10-s + 5.98·11-s − 3.49·13-s + 4.78i·14-s + 16-s − 4.96·17-s + 7.33i·19-s + (1.85 + 1.25i)20-s + 5.98·22-s + 1.74·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.829 + 0.559i)5-s + 1.81i·7-s + 0.353·8-s + (0.586 + 0.395i)10-s + 1.80·11-s − 0.969·13-s + 1.28i·14-s + 0.250·16-s − 1.20·17-s + 1.68i·19-s + (0.414 + 0.279i)20-s + 1.27·22-s + 0.364·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.487913549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.487913549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.85 - 1.25i)T \) |
| 37 | \( 1 + (3.96 + 4.61i)T \) |
| good | 7 | \( 1 - 4.78iT - 7T^{2} \) |
| 11 | \( 1 - 5.98T + 11T^{2} \) |
| 13 | \( 1 + 3.49T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 7.33iT - 19T^{2} \) |
| 23 | \( 1 - 1.74T + 23T^{2} \) |
| 29 | \( 1 + 7.85iT - 29T^{2} \) |
| 31 | \( 1 - 3.24iT - 31T^{2} \) |
| 41 | \( 1 - 0.530T + 41T^{2} \) |
| 43 | \( 1 - 1.76T + 43T^{2} \) |
| 47 | \( 1 + 4.30iT - 47T^{2} \) |
| 53 | \( 1 + 3.66iT - 53T^{2} \) |
| 59 | \( 1 - 2.15iT - 59T^{2} \) |
| 61 | \( 1 + 3.06iT - 61T^{2} \) |
| 67 | \( 1 - 3.79iT - 67T^{2} \) |
| 71 | \( 1 - 8.47T + 71T^{2} \) |
| 73 | \( 1 + 9.05iT - 73T^{2} \) |
| 79 | \( 1 + 5.56iT - 79T^{2} \) |
| 83 | \( 1 + 3.77iT - 83T^{2} \) |
| 89 | \( 1 - 8.45iT - 89T^{2} \) |
| 97 | \( 1 - 3.64T + 97T^{2} \) |
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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
−9.036638061776263980470759622655, −8.124863154335430136863391396344, −6.96776744627374435201897989400, −6.39359721507866295190285298031, −5.85829657860472392557404938677, −5.19250674506481419090609773292, −4.17363628164193361699370229640, −3.24134162536377317332962900769, −2.22186176257525086215590071402, −1.82936935063582624333291398405,
0.797996050584092551686557668841, 1.69110611784907131820014590725, 2.87744997288477787581140847562, 4.00334177611274540497942415640, 4.53798508808809286269710194472, 5.09810662413610495465353304618, 6.38130905418458751735780228478, 6.87590863038238011482516348027, 7.22978139316492817181052545863, 8.533218384904977320665674626979
