L(s) = 1 | + (2.12 − 0.707i)5-s − 5.65i·11-s + (−3 − 3i)13-s − 4i·19-s + (−2.82 + 2.82i)23-s + (3.99 − 3i)25-s + 1.41·29-s − 8·31-s + (−7 + 7i)37-s − 1.41i·41-s + (−4 − 4i)43-s + (−2.82 − 2.82i)47-s + 7i·49-s + (−8.48 + 8.48i)53-s + (−4.00 − 12i)55-s + ⋯ |
L(s) = 1 | + (0.948 − 0.316i)5-s − 1.70i·11-s + (−0.832 − 0.832i)13-s − 0.917i·19-s + (−0.589 + 0.589i)23-s + (0.799 − 0.600i)25-s + 0.262·29-s − 1.43·31-s + (−1.15 + 1.15i)37-s − 0.220i·41-s + (−0.609 − 0.609i)43-s + (−0.412 − 0.412i)47-s + i·49-s + (−1.16 + 1.16i)53-s + (−0.539 − 1.61i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.297668609\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297668609\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (7 - 7i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + (4 + 4i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.82 + 2.82i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.48 - 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (3 + 3i)T + 73iT^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 5i)T - 97iT^{2} \) |
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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
−8.579439092288309152090516584486, −7.83766549543443364303722333361, −6.88912266892462327946331583256, −6.04875020032366835358665743845, −5.44313364434606801545124964154, −4.82114643986369822028215117320, −3.46836789970035305268195261386, −2.79194143386596539272602371418, −1.62899641668849459845598898607, −0.37184448972186649717697009121,
1.88353560091855556645029996084, 2.06373058771165965089416450639, 3.45751092289754869993124149199, 4.50413201931727212126862812475, 5.15199038095200243812110491781, 6.06290941279953202443254619256, 6.93179376084952679061625924362, 7.28632171107719800268339154277, 8.342555142346870465979318118390, 9.283392584506873701152439271063