L(s) = 1 | − 1.41·7-s + (−1 − i)13-s + (−1.41 + 1.41i)19-s + i·25-s + 1.41i·31-s + (−1 + i)37-s + (−1.41 − 1.41i)43-s + 1.00·49-s + (−1 − i)61-s − 1.41i·79-s + (1.41 + 1.41i)91-s − 1.41·103-s + (1 + i)109-s + ⋯ |
L(s) = 1 | − 1.41·7-s + (−1 − i)13-s + (−1.41 + 1.41i)19-s + i·25-s + 1.41i·31-s + (−1 + i)37-s + (−1.41 − 1.41i)43-s + 1.00·49-s + (−1 − i)61-s − 1.41i·79-s + (1.41 + 1.41i)91-s − 1.41·103-s + (1 + i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1779741618\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1779741618\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - iT^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (1 + i)T + iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.618292646854421786414355708991, −8.763689757247044861762314383832, −8.039296437814855261216236881972, −7.08987441126375181163693687388, −6.49753798095318609385682754914, −5.65482761314676675087998370801, −4.85130223293669523379985459791, −3.58299271966217620820163601233, −3.10820471681110622902501709595, −1.82891550943925839986625081560,
0.10988874159758579008508723890, 2.15809311583789176247912629723, 2.88873767217867023359194083431, 4.07469871901779688092241481604, 4.69494694504989562345847928257, 5.89552621973001947210285539648, 6.67835761935869480428379474560, 7.01679888325089466211047989219, 8.148143711572473881557204080634, 9.054633312872521515384438179266