| L(s) = 1 | + 2-s + 4-s + 3.41·5-s − 2.87·7-s + 8-s + 3.41·10-s + 0.347·11-s − 1.65·13-s − 2.87·14-s + 16-s − 6.94·17-s + 3.41·20-s + 0.347·22-s − 6.80·23-s + 6.63·25-s − 1.65·26-s − 2.87·28-s − 6.35·29-s + 1.59·31-s + 32-s − 6.94·34-s − 9.82·35-s − 11.2·37-s + 3.41·40-s + 3.49·41-s + 2.28·43-s + 0.347·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.52·5-s − 1.08·7-s + 0.353·8-s + 1.07·10-s + 0.104·11-s − 0.458·13-s − 0.769·14-s + 0.250·16-s − 1.68·17-s + 0.762·20-s + 0.0740·22-s − 1.41·23-s + 1.32·25-s − 0.324·26-s − 0.544·28-s − 1.18·29-s + 0.286·31-s + 0.176·32-s − 1.19·34-s − 1.66·35-s − 1.84·37-s + 0.539·40-s + 0.545·41-s + 0.348·43-s + 0.0523·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 0.347T + 11T^{2} \) |
| 13 | \( 1 + 1.65T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 23 | \( 1 + 6.80T + 23T^{2} \) |
| 29 | \( 1 + 6.35T + 29T^{2} \) |
| 31 | \( 1 - 1.59T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 3.49T + 41T^{2} \) |
| 43 | \( 1 - 2.28T + 43T^{2} \) |
| 47 | \( 1 + 5.59T + 47T^{2} \) |
| 53 | \( 1 + 1.98T + 53T^{2} \) |
| 59 | \( 1 + 0.445T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 1.79T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−7.36175637080225246714215449973, −6.62090354728999973853549697178, −6.25605532102457011666658755744, −5.63212214525210787538948007173, −4.87824839256307397200098803215, −4.02046845199837694308594928894, −3.15725140358349786911134158473, −2.25256067298474632066857091127, −1.79383375507231470628226401712, 0,
1.79383375507231470628226401712, 2.25256067298474632066857091127, 3.15725140358349786911134158473, 4.02046845199837694308594928894, 4.87824839256307397200098803215, 5.63212214525210787538948007173, 6.25605532102457011666658755744, 6.62090354728999973853549697178, 7.36175637080225246714215449973
