L(s) = 1 | − 0.193·2-s − 1.96·4-s − 5-s − 3.35·7-s + 0.768·8-s + 0.193·10-s − 2.96·13-s + 0.649·14-s + 3.77·16-s − 4.57·17-s + 4.31·19-s + 1.96·20-s + 6.70·23-s + 25-s + 0.574·26-s + 6.57·28-s − 3.61·29-s + 9.92·31-s − 2.26·32-s + 0.887·34-s + 3.35·35-s − 2·37-s − 0.836·38-s − 0.768·40-s − 4.38·41-s + 9.27·43-s − 1.29·46-s + ⋯ |
L(s) = 1 | − 0.137·2-s − 0.981·4-s − 0.447·5-s − 1.26·7-s + 0.271·8-s + 0.0613·10-s − 0.821·13-s + 0.173·14-s + 0.943·16-s − 1.10·17-s + 0.989·19-s + 0.438·20-s + 1.39·23-s + 0.200·25-s + 0.112·26-s + 1.24·28-s − 0.670·29-s + 1.78·31-s − 0.401·32-s + 0.152·34-s + 0.566·35-s − 0.328·37-s − 0.135·38-s − 0.121·40-s − 0.685·41-s + 1.41·43-s − 0.191·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 17 | \( 1 + 4.57T + 17T^{2} \) |
| 19 | \( 1 - 4.31T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 4.38T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 - 9.92T + 47T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 - 5.92T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 - 0.0752T + 97T^{2} \) |
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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
−7.79095076085778356567920473831, −7.14313796504668757166890309028, −6.49581206103246332263642962859, −5.54271166514150688890944391012, −4.80961646063123247639063662300, −4.14288684155752306807239951676, −3.27942111342585387335074407755, −2.60631314058696451192810523850, −0.968584791953022547473595614955, 0,
0.968584791953022547473595614955, 2.60631314058696451192810523850, 3.27942111342585387335074407755, 4.14288684155752306807239951676, 4.80961646063123247639063662300, 5.54271166514150688890944391012, 6.49581206103246332263642962859, 7.14313796504668757166890309028, 7.79095076085778356567920473831