| L(s) = 1 | + (−1.16 + 0.247i)2-s + (−1.18 − 1.26i)3-s + (−0.528 + 0.235i)4-s + (0.918 − 2.03i)5-s + (1.69 + 1.18i)6-s + (0.157 + 0.272i)7-s + (2.48 − 1.80i)8-s + (−0.205 + 2.99i)9-s + (−0.566 + 2.60i)10-s + (−2.82 + 0.601i)11-s + (0.922 + 0.390i)12-s + (−6.38 − 1.35i)13-s + (−0.251 − 0.279i)14-s + (−3.66 + 1.24i)15-s + (−1.67 + 1.86i)16-s + (0.794 − 0.577i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 + 0.175i)2-s + (−0.682 − 0.730i)3-s + (−0.264 + 0.117i)4-s + (0.410 − 0.911i)5-s + (0.690 + 0.483i)6-s + (0.0595 + 0.103i)7-s + (0.879 − 0.638i)8-s + (−0.0684 + 0.997i)9-s + (−0.179 + 0.823i)10-s + (−0.852 + 0.181i)11-s + (0.266 + 0.112i)12-s + (−1.77 − 0.376i)13-s + (−0.0671 − 0.0746i)14-s + (−0.946 + 0.321i)15-s + (−0.419 + 0.465i)16-s + (0.192 − 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00508833 + 0.144076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00508833 + 0.144076i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.18 + 1.26i)T \) |
| 5 | \( 1 + (-0.918 + 2.03i)T \) |
| good | 2 | \( 1 + (1.16 - 0.247i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.157 - 0.272i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.82 - 0.601i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (6.38 + 1.35i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.794 + 0.577i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (5.88 - 4.27i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.876 + 0.973i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.450 + 4.28i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.905 - 8.61i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.00738 - 0.0227i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (4.48 + 0.952i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (1.34 + 2.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.544 + 5.18i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (3.44 + 2.50i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (5.06 + 1.07i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.24 + 1.54i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.260 - 2.47i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-9.63 - 7.00i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.283 - 0.872i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.18 + 11.3i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (9.08 + 4.04i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.78 + 11.6i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.772 + 7.35i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.07277478507881444456343022215, −10.36196249813299097837311843232, −9.970117364121616836923195225996, −8.545160082568407436903044332979, −7.938022785437791264706778753271, −6.90623346048856639806053040945, −5.43890291424914945353224510886, −4.64444001049620372191596039973, −2.01969669324218830824315205462, −0.16146367930413292197339113739,
2.49544543280569964478458192081, 4.43006779824891678622376180493, 5.40070015451744110461547494779, 6.72691459234444329588848127499, 7.86742546822741529620595375317, 9.286092416978092162590088625873, 9.876770545305431243944123391728, 10.67300888370035578390609617817, 11.23280728503617656609062236079, 12.56020897013585000619761597895
