| L(s) = 1 | + (1.85 + 0.741i)2-s + (2.90 + 2.75i)4-s + (−1.04 + 5.95i)5-s + (−1.18 − 1.41i)7-s + (3.34 + 7.26i)8-s + (−6.36 + 10.2i)10-s + (14.9 − 2.63i)11-s + (−6.81 + 2.48i)13-s + (−1.15 − 3.50i)14-s + (0.820 + 15.9i)16-s + (−2.22 + 3.85i)17-s + (−30.1 + 17.4i)19-s + (−19.4 + 14.3i)20-s + (29.7 + 6.19i)22-s + (−8.19 + 9.77i)23-s + ⋯ |
| L(s) = 1 | + (0.928 + 0.370i)2-s + (0.725 + 0.688i)4-s + (−0.209 + 1.19i)5-s + (−0.169 − 0.201i)7-s + (0.417 + 0.908i)8-s + (−0.636 + 1.02i)10-s + (1.36 − 0.239i)11-s + (−0.524 + 0.190i)13-s + (−0.0824 − 0.250i)14-s + (0.0512 + 0.998i)16-s + (−0.130 + 0.226i)17-s + (−1.58 + 0.917i)19-s + (−0.972 + 0.718i)20-s + (1.35 + 0.281i)22-s + (−0.356 + 0.424i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.71471 + 2.10943i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.71471 + 2.10943i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.85 - 0.741i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.04 - 5.95i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (1.18 + 1.41i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-14.9 + 2.63i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (6.81 - 2.48i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (2.22 - 3.85i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (30.1 - 17.4i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (8.19 - 9.77i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-26.3 - 9.59i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-34.0 + 40.5i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-1.20 + 2.08i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-44.2 + 16.1i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-15.3 + 2.71i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (13.3 + 15.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 77.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (69.2 + 12.2i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-25.1 + 21.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (25.3 + 69.6i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-81.7 - 47.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (42.1 + 72.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-0.961 + 2.64i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (19.3 - 53.0i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (55.7 + 96.5i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.0 + 73.9i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−11.76362027219168033840177418997, −10.94688314702088056508741758934, −10.05349970162324206854455819075, −8.605968862647114540210697269092, −7.51112281512103695796494473111, −6.57593257559572441060368005875, −6.08439751995434774052481286404, −4.34704655695964812275820641353, −3.58693911275268371938164035678, −2.25235472514766131460055593877,
1.01159275013001811877973848358, 2.58325849645721166302704691767, 4.27396403843425752007679191986, 4.70171111506523818466000958477, 6.09670335851546390274677672861, 6.94326824789387120805680604470, 8.474614131829690713694609377956, 9.280855057151114507870790785072, 10.34108741065008459200012165115, 11.45346362036141642725941790726
