| L(s) = 1 | + (−1.07 − 0.918i)2-s + (0.943 + 2.90i)3-s + (0.312 + 1.97i)4-s + (1.19 + 1.89i)5-s + (1.65 − 3.99i)6-s − 2.27i·7-s + (1.47 − 2.41i)8-s + (−5.11 + 3.71i)9-s + (0.456 − 3.12i)10-s + (−1.02 + 1.41i)11-s + (−5.44 + 2.77i)12-s + (−0.0225 + 0.0164i)13-s + (−2.09 + 2.44i)14-s + (−4.37 + 5.24i)15-s + (−3.80 + 1.23i)16-s + (4.95 + 1.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.760 − 0.649i)2-s + (0.544 + 1.67i)3-s + (0.156 + 0.987i)4-s + (0.533 + 0.846i)5-s + (0.674 − 1.62i)6-s − 0.860i·7-s + (0.522 − 0.852i)8-s + (−1.70 + 1.23i)9-s + (0.144 − 0.989i)10-s + (−0.309 + 0.425i)11-s + (−1.57 + 0.800i)12-s + (−0.00626 + 0.00454i)13-s + (−0.559 + 0.654i)14-s + (−1.12 + 1.35i)15-s + (−0.951 + 0.308i)16-s + (1.20 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.778563 + 0.649954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.778563 + 0.649954i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 + 0.918i)T \) |
| 5 | \( 1 + (-1.19 - 1.89i)T \) |
| good | 3 | \( 1 + (-0.943 - 2.90i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 2.27iT - 7T^{2} \) |
| 11 | \( 1 + (1.02 - 1.41i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.0225 - 0.0164i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.95 - 1.61i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (5.98 + 1.94i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.51 + 6.21i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-6.20 + 2.01i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.76 - 5.43i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.351 + 0.255i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.29 + 0.943i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 0.739T + 43T^{2} \) |
| 47 | \( 1 + (1.43 - 0.465i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (1.29 + 3.97i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.62 - 4.99i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.47 + 2.02i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.69 + 14.4i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.49 + 7.68i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.09 + 7.00i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (0.752 + 2.31i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.15 - 12.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-4.32 - 3.13i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (11.0 - 3.59i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.54646867809658256092315673418, −10.94447903828798965492918253068, −10.51730203854607832068843419090, −10.02958175599281857625290298741, −9.039861864843928061013882060026, −8.035072624740931138996358783804, −6.70148295745753582961406552541, −4.74658210472256536699249424148, −3.63894835814095464923861548801, −2.58944234722575780259284856426,
1.19012365716908282237649270325, 2.46969092914854481848691876372, 5.43961581705878459232028998098, 6.13013782460005672205923432407, 7.32733393009757227898460529744, 8.302785471796958850267920382220, 8.787205694058695998750193502664, 9.845736189701257462595944664265, 11.46871017528356421323775847798, 12.46641125060564094498988048200
