| L(s) = 1 | − 2.21·2-s − 3.14·3-s + 2.91·4-s − 5-s + 6.97·6-s + 7-s − 2.01·8-s + 6.89·9-s + 2.21·10-s − 0.929·11-s − 9.15·12-s − 3.79·13-s − 2.21·14-s + 3.14·15-s − 1.35·16-s + 4.22·17-s − 15.2·18-s + 6.05·19-s − 2.91·20-s − 3.14·21-s + 2.05·22-s − 2.31·23-s + 6.34·24-s + 25-s + 8.40·26-s − 12.2·27-s + 2.91·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 1.81·3-s + 1.45·4-s − 0.447·5-s + 2.84·6-s + 0.377·7-s − 0.713·8-s + 2.29·9-s + 0.700·10-s − 0.280·11-s − 2.64·12-s − 1.05·13-s − 0.592·14-s + 0.812·15-s − 0.337·16-s + 1.02·17-s − 3.60·18-s + 1.38·19-s − 0.650·20-s − 0.686·21-s + 0.439·22-s − 0.483·23-s + 1.29·24-s + 0.200·25-s + 1.64·26-s − 2.35·27-s + 0.549·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 3 | \( 1 + 3.14T + 3T^{2} \) |
| 11 | \( 1 + 0.929T + 11T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 + 2.31T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 0.815T + 31T^{2} \) |
| 37 | \( 1 + 3.25T + 37T^{2} \) |
| 41 | \( 1 + 0.482T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 + 3.95T + 47T^{2} \) |
| 53 | \( 1 + 0.427T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 - 2.67T + 61T^{2} \) |
| 67 | \( 1 + 8.28T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 1.55T + 79T^{2} \) |
| 83 | \( 1 - 9.54T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.53754819437867305429589620639, −7.05843533320737590673895746953, −6.27222924400937595439352476033, −5.41592311184856744419775773821, −4.96750565607150929932284331259, −4.12245495924603834867663605969, −2.81868956243339866616566011341, −1.57608150160103189277755370409, −0.869663066223122386674234095179, 0,
0.869663066223122386674234095179, 1.57608150160103189277755370409, 2.81868956243339866616566011341, 4.12245495924603834867663605969, 4.96750565607150929932284331259, 5.41592311184856744419775773821, 6.27222924400937595439352476033, 7.05843533320737590673895746953, 7.53754819437867305429589620639
