L(s) = 1 | + (2.84 + 0.938i)3-s − 6.81·7-s + (7.23 + 5.34i)9-s + 7.52i·11-s + 16.2·13-s + 4.11i·17-s − 7.86·19-s + (−19.4 − 6.39i)21-s + 19.5i·23-s + (15.6 + 22.0i)27-s + 55.8i·29-s + 43.4·31-s + (−7.06 + 21.4i)33-s + 31.5·37-s + (46.2 + 15.2i)39-s + ⋯ |
L(s) = 1 | + (0.949 + 0.312i)3-s − 0.973·7-s + (0.804 + 0.594i)9-s + 0.684i·11-s + 1.24·13-s + 0.242i·17-s − 0.413·19-s + (−0.924 − 0.304i)21-s + 0.848i·23-s + (0.578 + 0.815i)27-s + 1.92i·29-s + 1.40·31-s + (−0.214 + 0.650i)33-s + 0.853·37-s + (1.18 + 0.390i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.251856625\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.251856625\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.84 - 0.938i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 6.81T + 49T^{2} \) |
| 11 | \( 1 - 7.52iT - 121T^{2} \) |
| 13 | \( 1 - 16.2T + 169T^{2} \) |
| 17 | \( 1 - 4.11iT - 289T^{2} \) |
| 19 | \( 1 + 7.86T + 361T^{2} \) |
| 23 | \( 1 - 19.5iT - 529T^{2} \) |
| 29 | \( 1 - 55.8iT - 841T^{2} \) |
| 31 | \( 1 - 43.4T + 961T^{2} \) |
| 37 | \( 1 - 31.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 51.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 61.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 82.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 97.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 4.13T + 3.72e3T^{2} \) |
| 67 | \( 1 + 63.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 40.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 78.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 51.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 2.72iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 70.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44T + 9.40e3T^{2} \) |
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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
−10.32592568237174508228447884518, −9.783271645648740335298971620129, −8.873092917153352163825330932445, −8.200208977748352118943657114443, −7.07057299685539688052448688987, −6.28158534647080422993635142631, −4.89755673522430582909608111647, −3.76356721688023043751125748012, −3.01558640397469933023045900258, −1.56414329855897695279859393962,
0.795180008042637227741125184945, 2.46460922374427537065572978170, 3.40715764289670849074871799067, 4.34041439518733075192366165323, 6.14581583110322302870509924576, 6.50670301205730351550865350142, 7.85517506768592816866025874711, 8.489727685062063356626550629129, 9.334089158963752449478340732357, 10.09994777952513385202416407693