| L(s) = 1 | + (3.04 + 4.21i)3-s + (−4.90 + 8.49i)5-s + (13.1 + 13.0i)7-s + (−8.45 + 25.6i)9-s + (−0.742 + 0.428i)11-s + (−7.39 + 4.27i)13-s + (−50.7 + 5.21i)15-s + (47.9 − 83.0i)17-s + (−129. + 74.5i)19-s + (−15.1 + 95.0i)21-s + (30.7 + 17.7i)23-s + (14.3 + 24.8i)25-s + (−133. + 42.4i)27-s + (−76.3 − 44.0i)29-s + 39.0i·31-s + ⋯ |
| L(s) = 1 | + (0.586 + 0.810i)3-s + (−0.438 + 0.760i)5-s + (0.708 + 0.706i)7-s + (−0.313 + 0.949i)9-s + (−0.0203 + 0.0117i)11-s + (−0.157 + 0.0911i)13-s + (−0.873 + 0.0898i)15-s + (0.683 − 1.18i)17-s + (−1.55 + 0.900i)19-s + (−0.157 + 0.987i)21-s + (0.278 + 0.160i)23-s + (0.114 + 0.198i)25-s + (−0.953 + 0.302i)27-s + (−0.488 − 0.282i)29-s + 0.225i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.754268363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.754268363\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.04 - 4.21i)T \) |
| 7 | \( 1 + (-13.1 - 13.0i)T \) |
| good | 5 | \( 1 + (4.90 - 8.49i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (0.742 - 0.428i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (7.39 - 4.27i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-47.9 + 83.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (129. - 74.5i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-30.7 - 17.7i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (76.3 + 44.0i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 39.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (107. + 185. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-101. - 176. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (58.4 - 101. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 110.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (324. + 187. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 592.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 580. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 199.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 469. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-444. - 256. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 195.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-210. + 365. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (261. + 452. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-585. - 338. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.70711239524615559295720503560, −11.01314989922424378613565259242, −10.09558832026759251194682814842, −9.068050100864675283501597462471, −8.165364811043412244626318602779, −7.26215094999393482397124170362, −5.70947371281272215479508543096, −4.58920528423977629464539597924, −3.37236538992196302154771066156, −2.21572881233466442295601192935,
0.64213289543066037143553610727, 1.95577459925839226352955517408, 3.70217075467854519845526240481, 4.79999505256791138720398596160, 6.33907911121815261323256073454, 7.44638388286321094832056935959, 8.288892818100271043340874399838, 8.870930714488085301876587576170, 10.33512393116193390020164274312, 11.32114250889263476052243126544
