| L(s) = 1 | − 1.73i·3-s + 2.23i·5-s + (−6.13 + 3.36i)7-s − 2.99·9-s + 2.58·11-s + 0.0498i·13-s + 3.87·15-s − 14.2i·17-s + 14.9i·19-s + (5.82 + 10.6i)21-s + 22.2·23-s − 5.00·25-s + 5.19i·27-s + 17.4·29-s − 6.36i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s + (−0.877 + 0.480i)7-s − 0.333·9-s + 0.235·11-s + 0.00383i·13-s + 0.258·15-s − 0.835i·17-s + 0.785i·19-s + (0.277 + 0.506i)21-s + 0.968·23-s − 0.200·25-s + 0.192i·27-s + 0.600·29-s − 0.205i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.480i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4674795003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4674795003\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 + (6.13 - 3.36i)T \) |
| good | 11 | \( 1 - 2.58T + 121T^{2} \) |
| 13 | \( 1 - 0.0498iT - 169T^{2} \) |
| 17 | \( 1 + 14.2iT - 289T^{2} \) |
| 19 | \( 1 - 14.9iT - 361T^{2} \) |
| 23 | \( 1 - 22.2T + 529T^{2} \) |
| 29 | \( 1 - 17.4T + 841T^{2} \) |
| 31 | \( 1 + 6.36iT - 961T^{2} \) |
| 37 | \( 1 + 7.14T + 1.36e3T^{2} \) |
| 41 | \( 1 - 74.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 79.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 81.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 67.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.33iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 109. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 49.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 97.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 116. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 98.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 112. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109. iT - 9.40e3T^{2} \) |
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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
−8.774514137016871797216449435545, −8.058403173922793500974923589212, −7.00858964224528911741185165065, −6.59361741549884158757466563097, −5.74595101201108746600257631900, −4.79105662103403233330851742699, −3.40606542006564738183278397854, −2.83040975513436376799153562127, −1.60398512784772725357305637459, −0.13104080208839764914928281090,
1.20585741374104561937971404410, 2.76524226786878956038381465367, 3.65760340380165812492779254405, 4.47774265387992716715928721529, 5.32958956026806096167428182621, 6.32879443537869681244300089812, 6.99438293642303732010015979334, 8.026678647444090348413430218571, 8.960891886610361002549758168044, 9.350089252854910334530632810299
