| L(s) = 1 | + (25.7 + 2.90i)2-s + (82.6 + 51.9i)3-s + (404. + 92.3i)4-s + (418. − 333. i)5-s + (1.97e3 + 1.57e3i)6-s + (−324. − 1.42e3i)7-s + (3.88e3 + 1.35e3i)8-s + (1.28e3 + 2.66e3i)9-s + (1.17e4 − 7.37e3i)10-s + (−9.55e3 + 3.34e3i)11-s + (2.86e4 + 2.86e4i)12-s + (−1.42e4 + 2.96e4i)13-s + (−4.23e3 − 3.75e4i)14-s + (5.18e4 − 5.84e3i)15-s + (344. + 165. i)16-s + (−2.94e4 + 2.94e4i)17-s + ⋯ |
| L(s) = 1 | + (1.60 + 0.181i)2-s + (1.01 + 0.640i)3-s + (1.58 + 0.360i)4-s + (0.669 − 0.533i)5-s + (1.52 + 1.21i)6-s + (−0.135 − 0.592i)7-s + (0.948 + 0.331i)8-s + (0.195 + 0.406i)9-s + (1.17 − 0.737i)10-s + (−0.652 + 0.228i)11-s + (1.38 + 1.38i)12-s + (−0.500 + 1.03i)13-s + (−0.110 − 0.977i)14-s + (1.02 − 0.115i)15-s + (0.00525 + 0.00253i)16-s + (−0.352 + 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.52507 + 1.29823i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.52507 + 1.29823i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + (-6.66e5 - 2.37e5i)T \) |
| good | 2 | \( 1 + (-25.7 - 2.90i)T + (249. + 56.9i)T^{2} \) |
| 3 | \( 1 + (-82.6 - 51.9i)T + (2.84e3 + 5.91e3i)T^{2} \) |
| 5 | \( 1 + (-418. + 333. i)T + (8.69e4 - 3.80e5i)T^{2} \) |
| 7 | \( 1 + (324. + 1.42e3i)T + (-5.19e6 + 2.50e6i)T^{2} \) |
| 11 | \( 1 + (9.55e3 - 3.34e3i)T + (1.67e8 - 1.33e8i)T^{2} \) |
| 13 | \( 1 + (1.42e4 - 2.96e4i)T + (-5.08e8 - 6.37e8i)T^{2} \) |
| 17 | \( 1 + (2.94e4 - 2.94e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (2.15e4 + 3.43e4i)T + (-7.36e9 + 1.53e10i)T^{2} \) |
| 23 | \( 1 + (-2.96e4 + 3.71e4i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 31 | \( 1 + (-1.34e6 - 1.51e5i)T + (8.31e11 + 1.89e11i)T^{2} \) |
| 37 | \( 1 + (2.21e6 + 7.76e5i)T + (2.74e12 + 2.18e12i)T^{2} \) |
| 41 | \( 1 + (3.59e6 + 3.59e6i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (-8.03e4 - 7.12e5i)T + (-1.13e13 + 2.60e12i)T^{2} \) |
| 47 | \( 1 + (9.28e4 + 2.65e5i)T + (-1.86e13 + 1.48e13i)T^{2} \) |
| 53 | \( 1 + (-3.96e6 - 4.96e6i)T + (-1.38e13 + 6.06e13i)T^{2} \) |
| 59 | \( 1 - 1.68e7T + 1.46e14T^{2} \) |
| 61 | \( 1 + (-1.23e7 - 7.76e6i)T + (8.31e13 + 1.72e14i)T^{2} \) |
| 67 | \( 1 + (5.34e6 + 1.10e7i)T + (-2.53e14 + 3.17e14i)T^{2} \) |
| 71 | \( 1 + (1.53e7 - 3.17e7i)T + (-4.02e14 - 5.04e14i)T^{2} \) |
| 73 | \( 1 + (-2.20e7 + 2.47e6i)T + (7.86e14 - 1.79e14i)T^{2} \) |
| 79 | \( 1 + (-2.06e7 + 5.90e7i)T + (-1.18e15 - 9.45e14i)T^{2} \) |
| 83 | \( 1 + (1.89e7 - 8.28e7i)T + (-2.02e15 - 9.77e14i)T^{2} \) |
| 89 | \( 1 + (-8.03e7 - 9.05e6i)T + (3.83e15 + 8.75e14i)T^{2} \) |
| 97 | \( 1 + (2.18e7 - 1.37e7i)T + (3.40e15 - 7.06e15i)T^{2} \) |
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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
−15.06796529119044149796412604118, −13.97728313028616029730315419227, −13.43445505795332734467105319722, −12.11373378837404708634651805585, −10.20603287772904259395904237438, −8.826842883823393828677116916857, −6.81850044171727463458862629805, −5.05145761254740109291579933557, −3.93196485080538108314165547055, −2.40914830586388278222256459651,
2.33868592569179084583771527839, 3.00676178772587943597771967146, 5.22682861739245267588359714751, 6.59717706067594179232527437681, 8.284047002670367849367669911685, 10.26410868495136087386245694838, 11.98856021314456731047686208222, 13.16554894013831387526980514163, 13.76263123348190272613424521183, 14.76564133115058383538794029136
