| L(s) = 1 | + (−2.19 − 0.191i)2-s + (−1.58 + 0.699i)3-s + (2.79 + 0.492i)4-s + (−2.15 + 0.605i)5-s + (3.60 − 1.22i)6-s + (0.000530 + 0.000371i)7-s + (−1.77 − 0.474i)8-s + (2.02 − 2.21i)9-s + (4.83 − 0.913i)10-s + (0.925 − 2.54i)11-s + (−4.76 + 1.17i)12-s + (−0.277 − 3.16i)13-s + (−0.00109 − 0.000915i)14-s + (2.98 − 2.46i)15-s + (−1.53 − 0.559i)16-s + (6.58 − 1.76i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.135i)2-s + (−0.914 + 0.403i)3-s + (1.39 + 0.246i)4-s + (−0.962 + 0.270i)5-s + (1.47 − 0.501i)6-s + (0.000200 + 0.000140i)7-s + (−0.626 − 0.167i)8-s + (0.673 − 0.738i)9-s + (1.52 − 0.288i)10-s + (0.278 − 0.766i)11-s + (−1.37 + 0.338i)12-s + (−0.0768 − 0.878i)13-s + (−0.000291 − 0.000244i)14-s + (0.771 − 0.636i)15-s + (−0.384 − 0.139i)16-s + (1.59 − 0.428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.248778 - 0.147178i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.248778 - 0.147178i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.58 - 0.699i)T \) |
| 5 | \( 1 + (2.15 - 0.605i)T \) |
| good | 2 | \( 1 + (2.19 + 0.191i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.000530 - 0.000371i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.925 + 2.54i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.277 + 3.16i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-6.58 + 1.76i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.52 + 0.879i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.93 + 4.18i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (4.37 - 3.67i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.944 - 5.35i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.0678 + 0.253i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-7.56 + 9.01i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (3.67 + 7.88i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.403 - 0.576i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-5.73 + 5.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.11 - 0.768i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.14 + 6.48i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.47 + 0.216i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (10.4 + 6.02i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.375 - 1.40i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.36 - 4.00i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.39 - 15.9i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-2.99 - 5.18i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.62 + 3.55i)T + (62.3 - 74.3i)T^{2} \) |
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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
−12.43797179475666648155440824153, −11.67472615820404362389182410305, −10.75540930904058382925003581893, −10.16695055950743656189744329062, −8.933568941166225127782651213456, −7.85404471416823690530579512783, −6.92013514659745282927577989500, −5.36185205243734891411419215204, −3.49738801654892202414588033149, −0.61801080755678478012780944730,
1.40528473046258930421704853849, 4.33963944163376848807522668277, 6.08570409503337956461379899419, 7.53858656256695847078191654426, 7.74071908564556013013545618403, 9.343452604269430961868518719909, 10.15028560795921076496069294954, 11.44708886275760828999642367217, 11.85865954549471214612625182766, 13.04537031027645521382644405499
