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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.198997622\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.198997622\) |
\(L(1)\) |
|
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
−9.557446086959149255098683077100, −8.677190779699083676792159151011, −7.78606491713818434352256599561, −7.43955392502499907967682555885, −6.14222939154688672314160188700, −5.66702740947841132006920157471, −5.09662785367335967692893747719, −3.79051453753473926564416174066, −2.87467717383615755648803791688, −1.11350700702207569549548662926,
1.42032917235380558682707499699, 2.22472491269030609607377519732, 3.40488846095353589649218946965, 4.24795734900663932042724248736, 5.24864124594555686740093600968, 5.95868994977902670022733959138, 7.02632590247201524920917971935, 8.025248531824381133577420633695, 8.914185426021120442912112640840, 9.628001262722872473059309256848