| L(s) = 1 | − 8.11·5-s + 9.48i·7-s + 10.5i·11-s − 20.9·13-s − 4.92·17-s + 26.9i·19-s − 43.7i·23-s + 40.9·25-s − 14.5·29-s + 25.4i·31-s − 77.0i·35-s + 12.9·37-s + 50.1·41-s − 5.03i·43-s + 40.8i·47-s + ⋯ |
| L(s) = 1 | − 1.62·5-s + 1.35i·7-s + 0.962i·11-s − 1.61·13-s − 0.289·17-s + 1.41i·19-s − 1.90i·23-s + 1.63·25-s − 0.500·29-s + 0.822i·31-s − 2.20i·35-s + 0.350·37-s + 1.22·41-s − 0.117i·43-s + 0.869i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1535797366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1535797366\) |
| \(L(2)\) |
|
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 | \( 1 + 8.11T + 25T^{2} \) |
| 7 | \( 1 - 9.48iT - 49T^{2} \) |
| 11 | \( 1 - 10.5iT - 121T^{2} \) |
| 13 | \( 1 + 20.9T + 169T^{2} \) |
| 17 | \( 1 + 4.92T + 289T^{2} \) |
| 19 | \( 1 - 26.9iT - 361T^{2} \) |
| 23 | \( 1 + 43.7iT - 529T^{2} \) |
| 29 | \( 1 + 14.5T + 841T^{2} \) |
| 31 | \( 1 - 25.4iT - 961T^{2} \) |
| 37 | \( 1 - 12.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 50.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 5.03iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 27.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 28.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 127.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 15.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 73.8T + 9.40e3T^{2} \) |
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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
−9.314934915169722322266257396313, −8.476183246771512135521696989685, −7.77513099776829616160028450743, −7.12101120090098370833302712788, −6.04369822675492014686955747976, −4.84062070361736983544472357574, −4.33579835974944373938900567235, −3.03587143546808325986171727246, −2.10416955409844456043443902020, −0.06465180664600223555494092243,
0.806345565025790125134213096417, 2.78208811227511670578316613928, 3.80355348241842356032215426317, 4.38380893225313750016268314789, 5.39172351419640025175638262016, 6.87439997780340191424684193644, 7.46090873158675939729960387311, 7.83421766677603598675449943873, 8.969336631278642068320998247897, 9.807274246972435745057883440042
