| L(s) = 1 | − 4i·2-s − 16·4-s + 86.8i·5-s + 98.7i·7-s + 64i·8-s + 347.·10-s − 610. i·11-s + (592. − 143. i)13-s + 395.·14-s + 256·16-s + 1.14e3·17-s + 2.26e3i·19-s − 1.38e3i·20-s − 2.44e3·22-s − 433.·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.55i·5-s + 0.761i·7-s + 0.353i·8-s + 1.09·10-s − 1.52i·11-s + (0.971 − 0.234i)13-s + 0.538·14-s + 0.250·16-s + 0.963·17-s + 1.44i·19-s − 0.776i·20-s − 1.07·22-s − 0.170·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.261933762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261933762\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-592. + 143. i)T \) |
| good | 5 | \( 1 - 86.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 98.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 610. iT - 1.61e5T^{2} \) |
| 17 | \( 1 - 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.26e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 433.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.36e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.05e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 3.69e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 6.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.74e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.47e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.19e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 4.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.56e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.38e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 3.77e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.14e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.90e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 4.96e3iT - 8.58e9T^{2} \) |
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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
−11.39016413353933012732552657845, −10.76308597445775540388085946921, −9.962706446220888200175893222321, −8.688801069555330959361953351428, −7.80722662140288644923417936139, −6.26653386199517144480620083824, −5.62665225768673021523670576017, −3.47281876708367259058623711348, −3.13724758851073491568825200625, −1.54004811445409200498052538396,
0.37921355488310683196211631948, 1.59883081265575553430319927898, 3.99004334248554720880488743533, 4.73139053089582003258059214155, 5.78304801948556153484697870429, 7.19094174652952507656679588523, 7.918602933113753464778722624617, 9.145200425145832555467588288558, 9.612565490119138772806942403494, 11.02863556653891339517209650889
