| L(s) = 1 | − i·2-s + i·3-s − 4-s − 1.18i·5-s + 6-s + (1.74 + 1.99i)7-s + i·8-s − 9-s − 1.18·10-s + (−2.25 + 2.43i)11-s − i·12-s + 5.98·13-s + (1.99 − 1.74i)14-s + 1.18·15-s + 16-s + 4.29·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.530i·5-s + 0.408·6-s + (0.658 + 0.752i)7-s + 0.353i·8-s − 0.333·9-s − 0.374·10-s + (−0.678 + 0.734i)11-s − 0.288i·12-s + 1.65·13-s + (0.532 − 0.465i)14-s + 0.306·15-s + 0.250·16-s + 1.04·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44979 - 0.0198598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44979 - 0.0198598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (-1.74 - 1.99i)T \) |
| 11 | \( 1 + (2.25 - 2.43i)T \) |
| good | 5 | \( 1 + 1.18iT - 5T^{2} \) |
| 13 | \( 1 - 5.98T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 + 0.203T + 19T^{2} \) |
| 23 | \( 1 + 3.11T + 23T^{2} \) |
| 29 | \( 1 - 2.50iT - 29T^{2} \) |
| 31 | \( 1 - 2.79iT - 31T^{2} \) |
| 37 | \( 1 - 8.96T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 4.37iT - 43T^{2} \) |
| 47 | \( 1 + 2.05iT - 47T^{2} \) |
| 53 | \( 1 - 1.51T + 53T^{2} \) |
| 59 | \( 1 + 7.96iT - 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 2.29T + 73T^{2} \) |
| 79 | \( 1 - 9.85iT - 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 8.87iT - 89T^{2} \) |
| 97 | \( 1 - 9.46iT - 97T^{2} \) |
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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
−10.96599631271836121205575938646, −10.30581326831641325007177257619, −9.260190716372672492411190006178, −8.575873403722294330637112842843, −7.78293657489854765665525362203, −5.98029891835944966891410032682, −5.14605005773880659148799376651, −4.22426712016019197813510164022, −2.96684658634012975235248887970, −1.48964594292846673242596563019,
1.11290052069359868432660902566, 3.09580192147977834343612461057, 4.30227792250577007269576238407, 5.75651279352147987796689753707, 6.31594284167613288812799066773, 7.61651662891273986837866972816, 7.951064789402629153616604272449, 8.955530250363678047996601563240, 10.34562891175280478903778055984, 10.95485967137657614178909174671
