L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 7-s + 0.999·8-s + (−0.499 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)19-s + 0.999·20-s + 0.999·22-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 7-s + 0.999·8-s + (−0.499 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)19-s + 0.999·20-s + 0.999·22-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6794545469\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6794545469\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 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
−8.844047786716949057066935965556, −8.537304255024880709022964785813, −7.46268586290582438794133422486, −6.95583966826807034436603210237, −6.19738271677252500345239190107, −5.66026845929300098782881070959, −4.44901820758347014391162706101, −3.47077487476381426223089574212, −2.64355183091304648818436232957, −0.71763195939815427300915388734,
0.980925029934685749612827896588, 2.15443909938462274359583727140, 3.44213604577516428566122197221, 3.87246326021413010558483355856, 4.93174851592307173909917297673, 5.78471624221517926152583996373, 7.01812159067689128591763698961, 7.71365080051732131397662895713, 8.381583803784434653284213351961, 9.272230059152160766884585327618