L(s) = 1 | + (−1.58 − 1.58i)5-s + (3.23 − 3.23i)7-s + 4.57i·11-s + (4.23 + 4.23i)13-s + (1.74 + 1.74i)17-s + 2.47i·19-s + (−2.82 + 2.82i)23-s + 5.00i·25-s − 5.99·29-s − 1.52·31-s − 10.2·35-s + (−2.23 + 2.23i)37-s + 7.07i·41-s + (2.47 + 2.47i)43-s + (1.74 + 1.74i)47-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)5-s + (1.22 − 1.22i)7-s + 1.37i·11-s + (1.17 + 1.17i)13-s + (0.423 + 0.423i)17-s + 0.567i·19-s + (−0.589 + 0.589i)23-s + 1.00i·25-s − 1.11·29-s − 0.274·31-s − 1.72·35-s + (−0.367 + 0.367i)37-s + 1.10i·41-s + (0.376 + 0.376i)43-s + (0.254 + 0.254i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.668683111\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.668683111\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.58 + 1.58i)T \) |
good | 7 | \( 1 + (-3.23 + 3.23i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.57iT - 11T^{2} \) |
| 13 | \( 1 + (-4.23 - 4.23i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.74 - 1.74i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.47iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + (2.23 - 2.23i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-2.47 - 2.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.74 - 1.74i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + (1.52 - 1.52i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (0.527 + 0.527i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-1.08 + 1.08i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.746T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−8.757754521119951950543614402193, −7.950005007198946743228734518747, −7.56981997412818796096894183747, −6.82624523126416837825941355623, −5.70303284625830066699220213716, −4.71182398695653103047811252634, −4.17755444179425886926480193028, −3.69321393519520589648369205491, −1.71992530552276500403171971976, −1.33658893301678016102595327149,
0.57103952467319654110323522041, 2.05567241791152825441181239584, 3.07473211500504014950693770078, 3.68570294868216169972006185940, 4.88453455244419967941405118874, 5.76413563275445935277661356301, 6.09347234216423380208044007278, 7.43223108093251782223580462204, 7.946654310503484306530251508720, 8.666137940246570938216308357379