| L(s) = 1 | + (−0.118 + 0.822i)2-s + (1.08 + 1.24i)3-s + (1.25 + 0.368i)4-s + (−3.36 − 2.16i)5-s + (−1.15 + 0.742i)6-s + (0.0592 − 0.411i)7-s + (−1.14 + 2.50i)8-s + (0.0379 − 0.263i)9-s + (2.17 − 2.51i)10-s + (2.37 + 1.52i)11-s + (0.899 + 1.96i)12-s + (−1.87 − 4.11i)13-s + (0.331 + 0.0973i)14-s + (−0.942 − 6.55i)15-s + (0.281 + 0.181i)16-s + (−2.98 + 0.875i)17-s + ⋯ |
| L(s) = 1 | + (−0.0836 + 0.581i)2-s + (0.625 + 0.721i)3-s + (0.628 + 0.184i)4-s + (−1.50 − 0.967i)5-s + (−0.471 + 0.303i)6-s + (0.0223 − 0.155i)7-s + (−0.403 + 0.884i)8-s + (0.0126 − 0.0879i)9-s + (0.688 − 0.794i)10-s + (0.716 + 0.460i)11-s + (0.259 + 0.568i)12-s + (−0.521 − 1.14i)13-s + (0.0886 + 0.0260i)14-s + (−0.243 − 1.69i)15-s + (0.0704 + 0.0452i)16-s + (−0.723 + 0.212i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.856456 + 0.464542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.856456 + 0.464542i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 67 | \( 1 + (5.18 + 6.33i)T \) |
| good | 2 | \( 1 + (0.118 - 0.822i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-1.08 - 1.24i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (3.36 + 2.16i)T + (2.07 + 4.54i)T^{2} \) |
| 7 | \( 1 + (-0.0592 + 0.411i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-2.37 - 1.52i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.87 + 4.11i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.98 - 0.875i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.138 - 0.961i)T + (-18.2 + 5.35i)T^{2} \) |
| 23 | \( 1 + (2.58 + 2.98i)T + (-3.27 + 22.7i)T^{2} \) |
| 29 | \( 1 - 6.45T + 29T^{2} \) |
| 31 | \( 1 + (3.58 - 7.86i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + 4.42T + 37T^{2} \) |
| 41 | \( 1 + (9.39 - 2.75i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (3.36 - 0.988i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-2.58 - 2.98i)T + (-6.68 + 46.5i)T^{2} \) |
| 53 | \( 1 + (-6.68 - 1.96i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-6.00 + 13.1i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 0.917i)T + (25.3 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-3.30 - 0.969i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (1.54 - 0.992i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (2.41 + 5.27i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-5.37 - 3.45i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-3.33 + 3.84i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−15.34518242871794298332575567112, −14.47113380583759986185125119452, −12.48751404259640425880376283348, −11.91043716303719888346421058911, −10.43348007795043559226661942393, −8.796560244837191398718094400734, −8.141320977515605914627006337879, −6.88981900593092623587318670679, −4.79031463797246497598625192307, −3.47385461564117731945823248618,
2.37419043264165879908510815889, 3.82282857429771028782155094776, 6.74308471286334985002654447204, 7.34213396695948641444713185585, 8.728997002205128300219172773095, 10.43720911712625079864611521212, 11.69404877915624965454406798115, 11.86263552786708978910906515061, 13.60370484728038426461951908845, 14.68005537872742817784515271086
