| L(s) = 1 | + 2-s − 1.87·3-s + 4-s − 4.26·5-s − 1.87·6-s + 3.61·7-s + 8-s + 0.532·9-s − 4.26·10-s + 2.11·11-s − 1.87·12-s − 2.88·13-s + 3.61·14-s + 8.01·15-s + 16-s + 0.0267·17-s + 0.532·18-s − 3.22·19-s − 4.26·20-s − 6.78·21-s + 2.11·22-s + 2.97·23-s − 1.87·24-s + 13.1·25-s − 2.88·26-s + 4.63·27-s + 3.61·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.08·3-s + 0.5·4-s − 1.90·5-s − 0.767·6-s + 1.36·7-s + 0.353·8-s + 0.177·9-s − 1.34·10-s + 0.638·11-s − 0.542·12-s − 0.799·13-s + 0.965·14-s + 2.06·15-s + 0.250·16-s + 0.00648·17-s + 0.125·18-s − 0.740·19-s − 0.953·20-s − 1.48·21-s + 0.451·22-s + 0.619·23-s − 0.383·24-s + 2.63·25-s − 0.565·26-s + 0.892·27-s + 0.682·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 + 2.88T + 13T^{2} \) |
| 17 | \( 1 - 0.0267T + 17T^{2} \) |
| 19 | \( 1 + 3.22T + 19T^{2} \) |
| 23 | \( 1 - 2.97T + 23T^{2} \) |
| 29 | \( 1 - 5.73T + 29T^{2} \) |
| 31 | \( 1 + 6.76T + 31T^{2} \) |
| 41 | \( 1 + 1.49T + 41T^{2} \) |
| 43 | \( 1 + 5.53T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 - 1.90T + 53T^{2} \) |
| 59 | \( 1 + 5.62T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 + 9.27T + 71T^{2} \) |
| 73 | \( 1 + 16.2T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 1.69T + 83T^{2} \) |
| 89 | \( 1 - 8.27T + 89T^{2} \) |
| 97 | \( 1 - 6.52T + 97T^{2} \) |
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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
−8.303652494294730375942304116070, −7.45941707987259338321431554932, −7.00528167834033897086877166029, −6.04980210044688705607562972547, −4.93445140619642587493644670517, −4.70892326552684379816741671085, −3.94445583870229024359186907760, −2.88384500129424037395562121890, −1.34874604577150492113360889936, 0,
1.34874604577150492113360889936, 2.88384500129424037395562121890, 3.94445583870229024359186907760, 4.70892326552684379816741671085, 4.93445140619642587493644670517, 6.04980210044688705607562972547, 7.00528167834033897086877166029, 7.45941707987259338321431554932, 8.303652494294730375942304116070
