| L(s) = 1 | + (1.66 + 1.66i)5-s + 13.3i·7-s + (7.81 + 7.81i)11-s + (10.4 + 10.4i)13-s − 10.0i·17-s + (5.07 + 5.07i)19-s + 29.7·23-s − 19.4i·25-s + (−18.6 + 18.6i)29-s − 27.1·31-s + (−22.1 + 22.1i)35-s + (−13.3 + 13.3i)37-s − 34.9·41-s + (7.29 − 7.29i)43-s − 51.5i·47-s + ⋯ |
| L(s) = 1 | + (0.332 + 0.332i)5-s + 1.90i·7-s + (0.710 + 0.710i)11-s + (0.807 + 0.807i)13-s − 0.590i·17-s + (0.266 + 0.266i)19-s + 1.29·23-s − 0.778i·25-s + (−0.644 + 0.644i)29-s − 0.875·31-s + (−0.633 + 0.633i)35-s + (−0.360 + 0.360i)37-s − 0.852·41-s + (0.169 − 0.169i)43-s − 1.09i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.078762095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.078762095\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.66 - 1.66i)T + 25iT^{2} \) |
| 7 | \( 1 - 13.3iT - 49T^{2} \) |
| 11 | \( 1 + (-7.81 - 7.81i)T + 121iT^{2} \) |
| 13 | \( 1 + (-10.4 - 10.4i)T + 169iT^{2} \) |
| 17 | \( 1 + 10.0iT - 289T^{2} \) |
| 19 | \( 1 + (-5.07 - 5.07i)T + 361iT^{2} \) |
| 23 | \( 1 - 29.7T + 529T^{2} \) |
| 29 | \( 1 + (18.6 - 18.6i)T - 841iT^{2} \) |
| 31 | \( 1 + 27.1T + 961T^{2} \) |
| 37 | \( 1 + (13.3 - 13.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 34.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-7.29 + 7.29i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 51.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-68.4 - 68.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-31.5 - 31.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (72.6 + 72.6i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-60.3 - 60.3i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 3.09T + 5.04e3T^{2} \) |
| 73 | \( 1 + 2.05iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 53.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-21.7 + 21.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 137.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 17.1T + 9.40e3T^{2} \) |
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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
−9.613508261441966071919330478615, −9.024386956156760416653782856281, −8.580611821953842631607464180671, −7.19991402367060945133269998364, −6.50541730730966637709272222015, −5.65747467850625606945102384972, −4.88949633824036516911980782966, −3.57687152142067171521843701463, −2.48608001696671303941922138652, −1.61260555160344138947157376994,
0.66718981071345298089883856834, 1.45048385758260539350687480010, 3.37956391341591960654721167566, 3.86925682313810501584925599980, 5.03946354861390325452655751769, 5.99668890797621699096722761754, 6.92403297462419316225770719544, 7.61832659301420475511257072179, 8.556419679067673215373700272403, 9.348435200000731447319196462376
