| L(s) = 1 | − 2.35·2-s + 0.815·3-s + 3.54·4-s + 1.05·5-s − 1.92·6-s − 4.06·7-s − 3.64·8-s − 2.33·9-s − 2.49·10-s + 3.34·11-s + 2.89·12-s + 2.90·13-s + 9.57·14-s + 0.863·15-s + 1.48·16-s + 7.81·17-s + 5.49·18-s + 19-s + 3.75·20-s − 3.31·21-s − 7.88·22-s − 5.53·23-s − 2.97·24-s − 3.87·25-s − 6.83·26-s − 4.35·27-s − 14.4·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.470·3-s + 1.77·4-s + 0.473·5-s − 0.784·6-s − 1.53·7-s − 1.28·8-s − 0.778·9-s − 0.788·10-s + 1.00·11-s + 0.835·12-s + 0.804·13-s + 2.55·14-s + 0.222·15-s + 0.371·16-s + 1.89·17-s + 1.29·18-s + 0.229·19-s + 0.839·20-s − 0.723·21-s − 1.68·22-s − 1.15·23-s − 0.606·24-s − 0.775·25-s − 1.34·26-s − 0.837·27-s − 2.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 2 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 0.815T + 3T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 7.81T + 17T^{2} \) |
| 23 | \( 1 + 5.53T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 4.34T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 - 4.16T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 4.88T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 + 6.78T + 79T^{2} \) |
| 83 | \( 1 + 7.77T + 83T^{2} \) |
| 89 | \( 1 - 1.21T + 89T^{2} \) |
| 97 | \( 1 - 4.15T + 97T^{2} \) |
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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
−8.390285481102063626445816037974, −7.53388942372086871810083433331, −6.80121723369882774893460479291, −6.08509565937471853917888644476, −5.59589780509812620138177511561, −3.66098393692854954077790401917, −3.33713721844291076872814351718, −2.17875609708703332168121228325, −1.24777919890813495259151899640, 0,
1.24777919890813495259151899640, 2.17875609708703332168121228325, 3.33713721844291076872814351718, 3.66098393692854954077790401917, 5.59589780509812620138177511561, 6.08509565937471853917888644476, 6.80121723369882774893460479291, 7.53388942372086871810083433331, 8.390285481102063626445816037974
