| L(s) = 1 | + 2-s + 4-s − 5-s − 3.33·7-s + 8-s − 10-s − 11-s − 2.98·13-s − 3.33·14-s + 16-s + 4.19·17-s + 3.85·19-s − 20-s − 22-s + 8.56·23-s + 25-s − 2.98·26-s − 3.33·28-s + 3.24·29-s − 7.68·31-s + 32-s + 4.19·34-s + 3.33·35-s − 9.04·37-s + 3.85·38-s − 40-s − 6.64·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.26·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.826·13-s − 0.891·14-s + 0.250·16-s + 1.01·17-s + 0.883·19-s − 0.223·20-s − 0.213·22-s + 1.78·23-s + 0.200·25-s − 0.584·26-s − 0.630·28-s + 0.603·29-s − 1.38·31-s + 0.176·32-s + 0.719·34-s + 0.563·35-s − 1.48·37-s + 0.624·38-s − 0.158·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 3.33T + 7T^{2} \) |
| 13 | \( 1 + 2.98T + 13T^{2} \) |
| 17 | \( 1 - 4.19T + 17T^{2} \) |
| 19 | \( 1 - 3.85T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 - 3.24T + 29T^{2} \) |
| 31 | \( 1 + 7.68T + 31T^{2} \) |
| 37 | \( 1 + 9.04T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 - 4.73T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 8.75T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 + 5.98T + 71T^{2} \) |
| 73 | \( 1 + 2.89T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 + 9.72T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.16970377759148311809324187759, −6.89372753466907125570932985288, −5.91233263160729043686755701344, −5.25015853516094010865615628629, −4.75865806057292402302542120925, −3.54955377065429295227436771637, −3.29741931318220534184879085462, −2.55964349685779106894993466457, −1.25369946064891219968410421094, 0,
1.25369946064891219968410421094, 2.55964349685779106894993466457, 3.29741931318220534184879085462, 3.54955377065429295227436771637, 4.75865806057292402302542120925, 5.25015853516094010865615628629, 5.91233263160729043686755701344, 6.89372753466907125570932985288, 7.16970377759148311809324187759
