L(s) = 1 | + (2.11 + 0.726i)5-s + (3.61 − 3.61i)7-s − 3.26·11-s + (−3.37 + 3.37i)13-s + (−4.58 − 4.58i)17-s + 0.471·19-s + (3.32 − 3.32i)23-s + (3.94 + 3.07i)25-s − 5.98i·29-s + 5.98·31-s + (10.2 − 5.01i)35-s + (−0.639 − 0.639i)37-s − 3.39i·41-s + (3.08 − 3.08i)43-s + (−3.90 − 3.90i)47-s + ⋯ |
L(s) = 1 | + (0.945 + 0.324i)5-s + (1.36 − 1.36i)7-s − 0.984·11-s + (−0.937 + 0.937i)13-s + (−1.11 − 1.11i)17-s + 0.108·19-s + (0.693 − 0.693i)23-s + (0.788 + 0.614i)25-s − 1.11i·29-s + 1.07·31-s + (1.73 − 0.847i)35-s + (−0.105 − 0.105i)37-s − 0.530i·41-s + (0.470 − 0.470i)43-s + (−0.569 − 0.569i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.131713370\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.131713370\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.11 - 0.726i)T \) |
good | 7 | \( 1 + (-3.61 + 3.61i)T - 7iT^{2} \) |
| 11 | \( 1 + 3.26T + 11T^{2} \) |
| 13 | \( 1 + (3.37 - 3.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.58 + 4.58i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.471T + 19T^{2} \) |
| 23 | \( 1 + (-3.32 + 3.32i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.98iT - 29T^{2} \) |
| 31 | \( 1 - 5.98T + 31T^{2} \) |
| 37 | \( 1 + (0.639 + 0.639i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.39iT - 41T^{2} \) |
| 43 | \( 1 + (-3.08 + 3.08i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.90 + 3.90i)T + 47iT^{2} \) |
| 53 | \( 1 + (-1.26 - 1.26i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.2iT - 59T^{2} \) |
| 61 | \( 1 + 10.6iT - 61T^{2} \) |
| 67 | \( 1 + (-10.3 - 10.3i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.55iT - 71T^{2} \) |
| 73 | \( 1 + (-3.92 - 3.92i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.87iT - 79T^{2} \) |
| 83 | \( 1 + (-5.24 - 5.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + (-5.34 + 5.34i)T - 97iT^{2} \) |
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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
−8.540696574794544307239111581233, −7.81245547268814744102194135818, −6.98190807959519021353120853468, −6.68065507692579205180158422779, −5.20852679980798403718162462674, −4.87506993421944187408361221622, −4.07671095988765120089375263175, −2.57125357996087762026441791825, −2.04169501463680498831564122561, −0.66149664455381271219314363214,
1.39581117295178418849898607633, 2.30618024643272762988139178120, 2.89284523528127858240727983483, 4.64580785770309790794934581736, 5.10218511427431651791755639515, 5.65241722226703193729005604855, 6.46059694404378867777776926060, 7.68686677257590649213928428994, 8.202823138774539588355553827703, 8.893404714016338215991661582124