L(s) = 1 | + 1.10·2-s + (0.669 − 1.59i)3-s − 0.773·4-s − 1.62i·5-s + (0.741 − 1.76i)6-s + (2.53 + 0.762i)7-s − 3.07·8-s + (−2.10 − 2.13i)9-s − 1.80i·10-s + 2.37·11-s + (−0.517 + 1.23i)12-s + (2.05 − 2.96i)13-s + (2.80 + 0.844i)14-s + (−2.59 − 1.08i)15-s − 1.85·16-s − 5.91·17-s + ⋯ |
L(s) = 1 | + 0.783·2-s + (0.386 − 0.922i)3-s − 0.386·4-s − 0.727i·5-s + (0.302 − 0.722i)6-s + (0.957 + 0.288i)7-s − 1.08·8-s + (−0.701 − 0.712i)9-s − 0.569i·10-s + 0.715·11-s + (−0.149 + 0.356i)12-s + (0.570 − 0.821i)13-s + (0.749 + 0.225i)14-s + (−0.670 − 0.281i)15-s − 0.463·16-s − 1.43·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49897 - 1.13477i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49897 - 1.13477i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.669 + 1.59i)T \) |
| 7 | \( 1 + (-2.53 - 0.762i)T \) |
| 13 | \( 1 + (-2.05 + 2.96i)T \) |
good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 5 | \( 1 + 1.62iT - 5T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 7.30iT - 23T^{2} \) |
| 29 | \( 1 - 1.65iT - 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.27iT - 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 - 9.20iT - 47T^{2} \) |
| 53 | \( 1 + 12.3iT - 53T^{2} \) |
| 59 | \( 1 + 9.86iT - 59T^{2} \) |
| 61 | \( 1 + 2.47iT - 61T^{2} \) |
| 67 | \( 1 + 5.91iT - 67T^{2} \) |
| 71 | \( 1 + 8.43T + 71T^{2} \) |
| 73 | \( 1 + 7.77T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 5.44iT - 83T^{2} \) |
| 89 | \( 1 - 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 4.17T + 97T^{2} \) |
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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
−11.90120881168801098339970884785, −11.31234547886741217407777929732, −9.437847086293542990069955827939, −8.668469050131240789087233334694, −8.019920222419816131726860424020, −6.56172366183611926093916167079, −5.49738851363529275656260720100, −4.52663748220722164904053575334, −3.14837406237106339031033677874, −1.34458688415895151295402685976,
2.58668253072765851675599006846, 4.10542306148179770482639659174, 4.43764456940620397100718864440, 5.83295470112756470624198513155, 7.03116432642668460791558742705, 8.647339689770784538135811027749, 9.015580281160272611765077103730, 10.40072043469215830645710352396, 11.14260884634051554942110152471, 11.96512908607264489683748015524