L(s) = 1 | + (1.21 + 2.10i)3-s − 0.663·7-s + (−1.45 + 2.52i)9-s − 1.80·11-s + (−1.15 + 1.99i)13-s + (2.18 + 3.77i)17-s + (4.21 + 1.12i)19-s + (−0.806 − 1.39i)21-s + (−1.04 + 1.81i)23-s + 0.216·27-s + (−0.974 + 1.68i)29-s − 9.52·31-s + (−2.19 − 3.80i)33-s − 2.97·37-s − 5.60·39-s + ⋯ |
L(s) = 1 | + (0.701 + 1.21i)3-s − 0.250·7-s + (−0.485 + 0.840i)9-s − 0.545·11-s + (−0.319 + 0.553i)13-s + (0.528 + 0.915i)17-s + (0.966 + 0.257i)19-s + (−0.176 − 0.304i)21-s + (−0.218 + 0.378i)23-s + 0.0416·27-s + (−0.180 + 0.313i)29-s − 1.71·31-s + (−0.382 − 0.663i)33-s − 0.489·37-s − 0.896·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.603087092\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603087092\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-4.21 - 1.12i)T \) |
good | 3 | \( 1 + (-1.21 - 2.10i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 0.663T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + (1.15 - 1.99i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 3.77i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.04 - 1.81i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.974 - 1.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.52T + 31T^{2} \) |
| 37 | \( 1 + 2.97T + 37T^{2} \) |
| 41 | \( 1 + (0.247 + 0.428i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.93 - 6.81i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.28 - 5.69i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.15 - 1.99i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.88 + 6.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.36 - 9.28i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.29 + 3.96i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.95 + 5.12i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.80 + 4.86i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.99 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.20T + 83T^{2} \) |
| 89 | \( 1 + (-6.65 + 11.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.08 + 8.80i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.390243688039621978557835397692, −9.112221608717443027085027804170, −8.012143814197006536499336121536, −7.46563919993693154541061139606, −6.25255170852238253307568702270, −5.34596596789041187844360741781, −4.55936693591986889882191258007, −3.60770358549127678263991602040, −3.08312219900979702075891518995, −1.73843291023829082473626473155,
0.50960587103059906189934030604, 1.83808933556959734894152391541, 2.77634078885250290701435213947, 3.49447000999584816383576231628, 4.98504532156422374790693590650, 5.69901419551140304493055439728, 6.81737851696009222712365693006, 7.42423246184715226815886451081, 7.87646018830242356198009788179, 8.766157845329043816016820836244