L(s) = 1 | − 5.35i·2-s − 20.7·4-s + 4.43i·7-s + 68.1i·8-s + 3.43·11-s + 78.7i·13-s + 23.7·14-s + 199.·16-s + 53.1i·17-s − 20.4·19-s − 18.4i·22-s − 118. i·23-s + 421.·26-s − 91.8i·28-s + 168.·29-s + ⋯ |
L(s) = 1 | − 1.89i·2-s − 2.58·4-s + 0.239i·7-s + 3.01i·8-s + 0.0941·11-s + 1.67i·13-s + 0.453·14-s + 3.11·16-s + 0.758i·17-s − 0.246·19-s − 0.178i·22-s − 1.07i·23-s + 3.18·26-s − 0.620i·28-s + 1.07·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.949884 - 0.224237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.949884 - 0.224237i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 5.35iT - 8T^{2} \) |
| 7 | \( 1 - 4.43iT - 343T^{2} \) |
| 11 | \( 1 - 3.43T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 53.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 20.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 170. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 546. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 614. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 324.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.8iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 758. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 521iT - 9.12e5T^{2} \) |
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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
−11.81027719846880356459507140254, −10.83855705311154688492889704157, −10.04318153461359327077473369855, −9.058887260239526243841607637786, −8.368071932775210930947216354882, −6.48173484653766664869584686797, −4.81389652396368263983130818412, −3.95869801497453443313818730936, −2.57511246785799664603667742016, −1.41909186164965269806677811753,
0.44271855522955452617559092055, 3.53605949193784270753537104048, 4.97721268861969391748458586616, 5.72124477964164700179138551987, 6.90087421456586113881046906641, 7.70418956734380704437232279788, 8.538017321671135534020179315084, 9.577293405979104298973087674804, 10.55658960112043184698743327739, 12.23121114694583489255917825624