| L(s) = 1 | − 14·7-s − 3·9-s + 6·11-s + 6·23-s + 6·29-s + 110·37-s + 14·43-s + 147·49-s − 204·53-s + 42·63-s − 98·67-s + 246·71-s − 84·77-s + 94·79-s + 9·81-s − 18·99-s − 12·107-s − 178·109-s + 198·113-s − 215·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·7-s − 1/3·9-s + 6/11·11-s + 6/23·23-s + 6/29·29-s + 2.97·37-s + 0.325·43-s + 3·49-s − 3.84·53-s + 2/3·63-s − 1.46·67-s + 3.46·71-s − 1.09·77-s + 1.18·79-s + 1/9·81-s − 0.181·99-s − 0.112·107-s − 1.63·109-s + 1.75·113-s − 1.77·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.697712738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697712738\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 146 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 55 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2930 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 102 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 5234 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7010 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 123 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9890 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2978 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2350 T^{2} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.424657599793289091884702647468, −8.837418476918438893944987753606, −8.350456502630986893755759025539, −7.83108516133470912918458707039, −7.70652345656892935597584957183, −7.08019291181368337621852614160, −6.52959734784011808343240043981, −6.46751262888430805479165298255, −6.07802716435092484957323520762, −5.76938046508708261960155524971, −5.09197100269117472809614818542, −4.66099959578222565914818388859, −4.16308320416346394338981313519, −3.67244476410929665968230268052, −3.31437755124009445483328683766, −2.76825104371866181651003953368, −2.54535071745105602810182039267, −1.69958429860944584033395401808, −0.887435492793415097228662533986, −0.41077556690622847740239795635,
0.41077556690622847740239795635, 0.887435492793415097228662533986, 1.69958429860944584033395401808, 2.54535071745105602810182039267, 2.76825104371866181651003953368, 3.31437755124009445483328683766, 3.67244476410929665968230268052, 4.16308320416346394338981313519, 4.66099959578222565914818388859, 5.09197100269117472809614818542, 5.76938046508708261960155524971, 6.07802716435092484957323520762, 6.46751262888430805479165298255, 6.52959734784011808343240043981, 7.08019291181368337621852614160, 7.70652345656892935597584957183, 7.83108516133470912918458707039, 8.350456502630986893755759025539, 8.837418476918438893944987753606, 9.424657599793289091884702647468
