| L(s) = 1 | + (−27 − 241. i)3-s − 2.84e3i·5-s − 1.72e4·7-s + (−5.75e4 + 1.30e4i)9-s + 1.86e5i·11-s + 1.69e5·13-s + (−6.86e5 + 7.67e4i)15-s + 3.43e5i·17-s − 9.49e5·19-s + (4.65e5 + 4.16e6i)21-s + 2.65e6i·23-s + 1.67e6·25-s + (4.70e6 + 1.35e7i)27-s + 3.18e6i·29-s + 2.97e7·31-s + ⋯ |
| L(s) = 1 | + (−0.111 − 0.993i)3-s − 0.910i·5-s − 1.02·7-s + (−0.975 + 0.220i)9-s + 1.15i·11-s + 0.456·13-s + (−0.904 + 0.101i)15-s + 0.241i·17-s − 0.383·19-s + (0.113 + 1.01i)21-s + 0.412i·23-s + 0.171·25-s + (0.327 + 0.944i)27-s + 0.155i·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.454914522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.454914522\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (27 + 241. i)T \) |
| good | 5 | \( 1 + 2.84e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 1.72e4T + 2.82e8T^{2} \) |
| 11 | \( 1 - 1.86e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.69e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 3.43e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 9.49e5T + 6.13e12T^{2} \) |
| 23 | \( 1 - 2.65e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 3.18e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 2.97e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 6.08e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.81e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 1.07e8T + 2.16e16T^{2} \) |
| 47 | \( 1 + 2.67e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 1.92e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 6.49e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.03e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.87e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 2.68e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 2.84e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 1.48e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.26e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 6.02e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.59e9T + 7.37e19T^{2} \) |
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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
−10.43265657982116885781851658132, −9.320039142341693812978915200183, −8.473981379431487394709155453132, −7.34089621549535636383222070578, −6.45942310647907818795977808429, −5.44181124898794563649470293317, −4.15056635484960529175847844445, −2.68413580122820002328509169172, −1.51825321329571959784349352808, −0.50930151040333073525041922453,
0.62608703403506273921554213535, 2.81181935222612321665367213445, 3.29207424213232910450690762844, 4.50538380510861548404376285536, 6.02560410998960699396858131457, 6.47005040549149289560923756399, 8.086743137648584726443805130670, 9.149615692146227788061899445331, 10.02874162355735847847798366992, 10.86465490284613077628087283716
