| L(s) = 1 | − 1.53·2-s − 3-s + 0.347·4-s − 2.53·5-s + 1.53·6-s + 0.532·7-s + 2.53·8-s + 9-s + 3.87·10-s − 5.10·11-s − 0.347·12-s + 4.06·13-s − 0.815·14-s + 2.53·15-s − 4.57·16-s + 1.94·17-s − 1.53·18-s − 0.879·20-s − 0.532·21-s + 7.82·22-s + 3.04·23-s − 2.53·24-s + 1.41·25-s − 6.22·26-s − 27-s + 0.184·28-s + 1.61·29-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.577·3-s + 0.173·4-s − 1.13·5-s + 0.625·6-s + 0.201·7-s + 0.895·8-s + 0.333·9-s + 1.22·10-s − 1.53·11-s − 0.100·12-s + 1.12·13-s − 0.217·14-s + 0.653·15-s − 1.14·16-s + 0.471·17-s − 0.361·18-s − 0.196·20-s − 0.116·21-s + 1.66·22-s + 0.634·23-s − 0.516·24-s + 0.282·25-s − 1.22·26-s − 0.192·27-s + 0.0349·28-s + 0.299·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 - 4.06T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 - 1.61T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 + 8.47T + 41T^{2} \) |
| 43 | \( 1 + 0.177T + 43T^{2} \) |
| 47 | \( 1 - 7.55T + 47T^{2} \) |
| 53 | \( 1 + 9.90T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.82T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 5.92T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−9.488601167915191482284005119374, −8.349853554280412119126144311132, −8.044000765039110510589496319204, −7.33394245576966388956990435234, −6.21694718878838192904878275397, −5.02444409259101313378367072729, −4.31483958390509541490332932498, −3.01837483408698336714371223656, −1.23055308651576261087653280176, 0,
1.23055308651576261087653280176, 3.01837483408698336714371223656, 4.31483958390509541490332932498, 5.02444409259101313378367072729, 6.21694718878838192904878275397, 7.33394245576966388956990435234, 8.044000765039110510589496319204, 8.349853554280412119126144311132, 9.488601167915191482284005119374
