L(s) = 1 | + i·2-s − 4-s − 0.765·5-s − i·8-s − 0.765i·10-s + 1.84i·13-s + 16-s − 1.84·17-s + 0.765·20-s − 0.414·25-s − 1.84·26-s + i·32-s − 1.84i·34-s − 1.41·37-s + 0.765i·40-s − 1.84·41-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − 0.765·5-s − i·8-s − 0.765i·10-s + 1.84i·13-s + 16-s − 1.84·17-s + 0.765·20-s − 0.414·25-s − 1.84·26-s + i·32-s − 1.84i·34-s − 1.41·37-s + 0.765i·40-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3628726676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3628726676\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.765T + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.84iT - T^{2} \) |
| 17 | \( 1 + 1.84T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 + 1.84T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 0.765iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 0.765iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.765T + T^{2} \) |
| 97 | \( 1 + 0.765iT - T^{2} \) |
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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
−9.614227038473247761963582378482, −8.861690191637752517288200976965, −8.470874415411871964847949793160, −7.37286369535516147596222139474, −6.83554774700104499351194627973, −6.21164277243946646649480727585, −4.93481455176571975384976323156, −4.33909734910661958073156051152, −3.59292354269028910237545565392, −1.91706041128369932223437314601,
0.26632204724506199773764490625, 1.94654061436474676738336277594, 3.08092458944129551683799876104, 3.78761371303869051775689980247, 4.76634290584006806791108200997, 5.49227512797760681584148060602, 6.69297214205565871997125328914, 7.74916555843621085456337787344, 8.403656335657013308448349282119, 8.985573759870906334660058223607