| L(s) = 1 | + (1 + 1.73i)2-s − 1.73i·3-s + (−1.99 + 3.46i)4-s + (2.99 − 1.73i)6-s − 10.3i·7-s − 7.99·8-s − 2.99·9-s − 10.3i·11-s + (5.99 + 3.46i)12-s + 18·13-s + (18 − 10.3i)14-s + (−8 − 13.8i)16-s + 10·17-s + (−2.99 − 5.19i)18-s − 13.8i·19-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.866i)2-s − 0.577i·3-s + (−0.499 + 0.866i)4-s + (0.499 − 0.288i)6-s − 1.48i·7-s − 0.999·8-s − 0.333·9-s − 0.944i·11-s + (0.499 + 0.288i)12-s + 1.38·13-s + (1.28 − 0.742i)14-s + (−0.5 − 0.866i)16-s + 0.588·17-s + (−0.166 − 0.288i)18-s − 0.729i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.76711 - 0.473496i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76711 - 0.473496i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - 1.73i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 10.3iT - 49T^{2} \) |
| 11 | \( 1 + 10.3iT - 121T^{2} \) |
| 13 | \( 1 - 18T + 169T^{2} \) |
| 17 | \( 1 - 10T + 289T^{2} \) |
| 19 | \( 1 + 13.8iT - 361T^{2} \) |
| 23 | \( 1 + 6.92iT - 529T^{2} \) |
| 29 | \( 1 + 36T + 841T^{2} \) |
| 31 | \( 1 - 6.92iT - 961T^{2} \) |
| 37 | \( 1 - 54T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 26T + 2.80e3T^{2} \) |
| 59 | \( 1 - 31.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74T + 3.72e3T^{2} \) |
| 67 | \( 1 + 41.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 36T + 5.32e3T^{2} \) |
| 79 | \( 1 + 90.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 90.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 18T + 7.92e3T^{2} \) |
| 97 | \( 1 + 72T + 9.40e3T^{2} \) |
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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
−11.44641659161806970646949378119, −10.76020046130107952399319556028, −9.256410222527187871226993973719, −8.222119278958223583640863311540, −7.47795926774721656314499227414, −6.54122075787208288844131846417, −5.68917968192121087710075349027, −4.21367817798966557315072725753, −3.27355068860280451216097271988, −0.812249866827834641146803614656,
1.79625612887449332423523611722, 3.13068473769664397930115799277, 4.26911001390627901531722190622, 5.50416165200524974151139829966, 6.11657885591481747662234166419, 8.074054975502293242663106833871, 9.185844305479484554992814331839, 9.687317209334965565593538840097, 10.84868311889043295451243031892, 11.60234847954900137159105801100
