| L(s) = 1 | + 15.5i·3-s − 107. i·7-s − 243·9-s + 1.20e3·13-s + 2.80e3i·19-s + 1.67e3·21-s + 3.12e3·25-s − 3.78e3i·27-s + 2.81e3i·31-s − 1.65e4·37-s + 1.87e4i·39-s + 2.40e4i·43-s + 5.27e3·49-s − 4.36e4·57-s + 3.86e4·61-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s − 0.828i·7-s − 9-s + 1.97·13-s + 1.78i·19-s + 0.828·21-s + 25-s − 1.00i·27-s + 0.526i·31-s − 1.98·37-s + 1.97i·39-s + 1.98i·43-s + 0.313·49-s − 1.78·57-s + 1.32·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.867248391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.867248391\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5iT \) |
| good | 5 | \( 1 - 3.12e3T^{2} \) |
| 7 | \( 1 + 107. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.20e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.80e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.81e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.65e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.15e8T^{2} \) |
| 43 | \( 1 - 2.40e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.43e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.45e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.68e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.34e5T + 8.58e9T^{2} \) |
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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
−11.63320948851240841647611729683, −10.68415613373045265184274991948, −10.18872131841901903748712317135, −8.879290033560392649422035915079, −8.111066823814208726958604509033, −6.55987681479371000024761533393, −5.47950722604040903197899933399, −4.09157782044275270518137119568, −3.36950289415907175661937133932, −1.22518783736578990330557901001,
0.66476709479072142129475142476, 2.05652942633563333049894265028, 3.35949394079718829127442004073, 5.23191296068089890820530308938, 6.26996523397665687498222192207, 7.14856256120258757932112919024, 8.659706740944889884598262326530, 8.846486679487611342418607639077, 10.73373892615096661357399166337, 11.48423878379929271081370102813
