| L(s) = 1 | + 0.285i·3-s − 5.85·7-s + 8.91·9-s − 4.75i·11-s + 11.4i·13-s + 4.16·17-s − 21.4i·19-s − 1.67i·21-s + (−13.1 + 18.8i)23-s + 5.11i·27-s − 4.76·29-s − 2.69·31-s + 1.35·33-s + 49.8·37-s − 3.26·39-s + ⋯ |
| L(s) = 1 | + 0.0950i·3-s − 0.837·7-s + 0.990·9-s − 0.431i·11-s + 0.881i·13-s + 0.245·17-s − 1.12i·19-s − 0.0795i·21-s + (−0.571 + 0.820i)23-s + 0.189i·27-s − 0.164·29-s − 0.0869·31-s + 0.0410·33-s + 1.34·37-s − 0.0838·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.853526336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.853526336\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (13.1 - 18.8i)T \) |
| good | 3 | \( 1 - 0.285iT - 9T^{2} \) |
| 7 | \( 1 + 5.85T + 49T^{2} \) |
| 11 | \( 1 + 4.75iT - 121T^{2} \) |
| 13 | \( 1 - 11.4iT - 169T^{2} \) |
| 17 | \( 1 - 4.16T + 289T^{2} \) |
| 19 | \( 1 + 21.4iT - 361T^{2} \) |
| 29 | \( 1 + 4.76T + 841T^{2} \) |
| 31 | \( 1 + 2.69T + 961T^{2} \) |
| 37 | \( 1 - 49.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 15.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 18.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 29.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 38.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 77.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 48.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 13.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 15.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 52.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 52.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 37.6T + 9.40e3T^{2} \) |
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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
−9.270814098199775181945790900329, −8.037576116275167867977055916562, −7.27838048589325495909801971720, −6.58585063341210882341263355787, −5.88941859257588323840469398472, −4.78430720078702992483658066716, −4.03013993669030128821001563655, −3.17125201058590323278554002730, −2.05121203640233618606157549768, −0.808817206504120208200443561285,
0.62355172164044521722659840727, 1.85030617459553094926060551052, 2.96818084242508371563662922779, 3.89459422166308311105997957599, 4.65738464590534588607254819583, 5.83155942634481066984640488811, 6.33382303267492785037350907616, 7.37054598575213550951704770493, 7.80217905943750761615751066908, 8.791347148846834098866386836561
