L(s) = 1 | − 2-s + 4-s + (−0.866 + 1.5i)5-s − 8-s + (−0.5 − 0.866i)9-s + (0.866 − 1.5i)10-s + (0.866 − 0.5i)13-s + 16-s + 1.73·17-s + (0.5 + 0.866i)18-s + (−0.866 + 1.5i)20-s + (−1 − 1.73i)25-s + (−0.866 + 0.5i)26-s + (−0.5 − 0.866i)29-s − 32-s + ⋯ |
L(s) = 1 | − 2-s + 4-s + (−0.866 + 1.5i)5-s − 8-s + (−0.5 − 0.866i)9-s + (0.866 − 1.5i)10-s + (0.866 − 0.5i)13-s + 16-s + 1.73·17-s + (0.5 + 0.866i)18-s + (−0.866 + 1.5i)20-s + (−1 − 1.73i)25-s + (−0.866 + 0.5i)26-s + (−0.5 − 0.866i)29-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6725513551\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6725513551\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.73T + T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
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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
−9.209706524802917361065863481972, −8.175472448687450532603400116778, −7.77237561284497325615854678041, −7.09062962398852587392837143626, −6.13926808291233769193942062268, −5.85143774385543458376634041612, −3.95234506162404081256874999068, −3.23879715059415405639987190946, −2.68228059722905860880809645336, −0.969215570395975381207184495575,
0.859126016796920862000896399099, 1.84840851776263443178330801696, 3.25923894354767189059675998315, 4.14369051358395794536108755945, 5.32366294935638653495774643462, 5.74408626318930151809620209747, 7.06880365079133966081293620477, 7.81487848817717319221338939729, 8.285796655435949202884492955345, 8.846013986235679391774765432415