L(s) = 1 | + (2.87 − 0.845i)3-s + (2.32 + 3.25i)4-s + (−0.560 − 11.7i)7-s + (7.57 − 4.86i)9-s + (9.43 + 7.41i)12-s + (11.8 − 4.75i)13-s + (−5.23 + 15.1i)16-s + (−1.55 + 32.7i)19-s + (−11.5 − 33.3i)21-s + (−3.55 + 24.7i)25-s + (17.6 − 20.4i)27-s + (37.0 − 29.1i)28-s + (−34.6 − 13.8i)31-s + (33.4 + 13.3i)36-s + (−32.4 − 56.1i)37-s + ⋯ |
L(s) = 1 | + (0.959 − 0.281i)3-s + (0.580 + 0.814i)4-s + (−0.0800 − 1.68i)7-s + (0.841 − 0.540i)9-s + (0.786 + 0.618i)12-s + (0.913 − 0.365i)13-s + (−0.327 + 0.945i)16-s + (−0.0820 + 1.72i)19-s + (−0.550 − 1.58i)21-s + (−0.142 + 0.989i)25-s + (0.654 − 0.755i)27-s + (1.32 − 1.03i)28-s + (−1.11 − 0.447i)31-s + (0.928 + 0.371i)36-s + (−0.876 − 1.51i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.33281 - 0.293629i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.33281 - 0.293629i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.87 + 0.845i)T \) |
| 67 | \( 1 + (30.8 + 59.4i)T \) |
good | 2 | \( 1 + (-2.32 - 3.25i)T^{2} \) |
| 5 | \( 1 + (3.55 - 24.7i)T^{2} \) |
| 7 | \( 1 + (0.560 + 11.7i)T + (-48.7 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-112. + 44.9i)T^{2} \) |
| 13 | \( 1 + (-11.8 + 4.75i)T + (122. - 116. i)T^{2} \) |
| 17 | \( 1 + (94.5 + 273. i)T^{2} \) |
| 19 | \( 1 + (1.55 - 32.7i)T + (-359. - 34.3i)T^{2} \) |
| 23 | \( 1 + (-25.1 - 528. i)T^{2} \) |
| 29 | \( 1 + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (34.6 + 13.8i)T + (695. + 663. i)T^{2} \) |
| 37 | \( 1 + (32.4 + 56.1i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-1.65e3 - 318. i)T^{2} \) |
| 43 | \( 1 + (-28.5 - 62.4i)T + (-1.21e3 + 1.39e3i)T^{2} \) |
| 47 | \( 1 + (1.96e3 + 1.01e3i)T^{2} \) |
| 53 | \( 1 + (1.83e3 + 2.12e3i)T^{2} \) |
| 59 | \( 1 + (3.33e3 - 980. i)T^{2} \) |
| 61 | \( 1 + (75.4 + 14.5i)T + (3.45e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (1.64e3 - 4.76e3i)T^{2} \) |
| 73 | \( 1 + (113. + 21.9i)T + (4.94e3 + 1.98e3i)T^{2} \) |
| 79 | \( 1 + (-83.3 - 65.5i)T + (1.47e3 + 6.06e3i)T^{2} \) |
| 83 | \( 1 + (5.41e3 + 4.25e3i)T^{2} \) |
| 89 | \( 1 + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-66.2 - 114. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.48342129509224319043060978218, −11.09098431283006711250558395216, −10.32600544170124139462534786906, −9.043171278047047624751379213353, −7.75970222018742343583314179564, −7.51794270264040394100316822568, −6.26707157286055035371212530098, −3.95354146499585882055494628834, −3.44206758197312326612521681430, −1.58007225538203431605312158414,
1.93723293411883902598718991359, 2.97296439022538461235603547678, 4.82031736195853107531835613654, 5.98090892850158432772338509757, 7.08125447579306571491602458434, 8.689905931086949491800065316203, 9.047129936001625964525655117048, 10.21629671668754223134567078228, 11.23479821588937656823021995450, 12.18593186622698970201584358975