| L(s) = 1 | + 1.73i·3-s + 5.91i·5-s + (6.24 − 3.16i)7-s − 2.99·9-s + 1.75·11-s − 18.7i·13-s − 10.2·15-s + 23.4i·17-s + 23.0i·19-s + (5.48 + 10.8i)21-s + 18.7·23-s − 9.97·25-s − 5.19i·27-s − 30·29-s + 8.60i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.18i·5-s + (0.891 − 0.452i)7-s − 0.333·9-s + 0.159·11-s − 1.44i·13-s − 0.682·15-s + 1.38i·17-s + 1.21i·19-s + (0.261 + 0.514i)21-s + 0.814·23-s − 0.398·25-s − 0.192i·27-s − 1.03·29-s + 0.277i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.977297603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.977297603\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-6.24 + 3.16i)T \) |
| good | 5 | \( 1 - 5.91iT - 25T^{2} \) |
| 11 | \( 1 - 1.75T + 121T^{2} \) |
| 13 | \( 1 + 18.7iT - 169T^{2} \) |
| 17 | \( 1 - 23.4iT - 289T^{2} \) |
| 19 | \( 1 - 23.0iT - 361T^{2} \) |
| 23 | \( 1 - 18.7T + 529T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 - 8.60iT - 961T^{2} \) |
| 37 | \( 1 - 70.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 38.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 37.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 97.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 56.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 69.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 6.43iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 51.7iT - 9.40e3T^{2} \) |
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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
−10.02977969141084538825910312520, −8.830422736271953833469956000083, −7.903419231647006591944245118687, −7.49244213978709360445741908950, −6.22771619852616663425141414311, −5.63535255631316530582342974467, −4.45883750805388535130340446872, −3.60156119378857990452199575773, −2.75713366255799707092192601859, −1.32947588471373765406750353109,
0.60411187648636625674904132700, 1.66326669115255394424921806690, 2.63273712164987960096230158387, 4.30068351823485058071928243641, 4.90507752496870346460801064650, 5.67269697349685619656040317229, 6.89117895334402954895743396304, 7.46814867833688499440369389506, 8.552756829059890765972056094366, 9.046719548379831889737254661967
