| L(s) = 1 | + (1.41 + 0.0995i)2-s + (0.112 + 0.346i)3-s + (1.98 + 0.280i)4-s + (−1.22 − 1.86i)5-s + (0.124 + 0.499i)6-s + 4.30i·7-s + (2.76 + 0.593i)8-s + (2.31 − 1.68i)9-s + (−1.54 − 2.75i)10-s + (1.67 − 2.30i)11-s + (0.125 + 0.717i)12-s + (−1.64 + 1.19i)13-s + (−0.428 + 6.06i)14-s + (0.508 − 0.635i)15-s + (3.84 + 1.11i)16-s + (−6.70 − 2.17i)17-s + ⋯ |
| L(s) = 1 | + (0.997 + 0.0704i)2-s + (0.0649 + 0.199i)3-s + (0.990 + 0.140i)4-s + (−0.549 − 0.835i)5-s + (0.0507 + 0.204i)6-s + 1.62i·7-s + (0.977 + 0.209i)8-s + (0.773 − 0.561i)9-s + (−0.489 − 0.872i)10-s + (0.504 − 0.694i)11-s + (0.0362 + 0.207i)12-s + (−0.455 + 0.330i)13-s + (−0.114 + 1.62i)14-s + (0.131 − 0.164i)15-s + (0.960 + 0.278i)16-s + (−1.62 − 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01670 + 0.200780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01670 + 0.200780i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0995i)T \) |
| 5 | \( 1 + (1.22 + 1.86i)T \) |
| good | 3 | \( 1 + (-0.112 - 0.346i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 4.30iT - 7T^{2} \) |
| 11 | \( 1 + (-1.67 + 2.30i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (1.64 - 1.19i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.70 + 2.17i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (5.54 + 1.80i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.945 + 1.30i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (7.60 - 2.47i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.685 - 2.10i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.672 + 0.488i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-7.65 + 5.56i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.08T + 43T^{2} \) |
| 47 | \( 1 + (0.833 - 0.270i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.578 + 1.78i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.01 - 11.0i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.59 + 6.33i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (1.35 - 4.16i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.71 + 5.28i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.126 + 0.174i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (1.86 + 5.73i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (2.65 - 8.17i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.97 - 1.43i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.16 + 1.67i)T + (78.4 - 57.0i)T^{2} \) |
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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
−12.61498426830711158284718764191, −11.75639390655199864335306839435, −11.01846476931226161942708774459, −9.170853578516675166935140474684, −8.730220458502459227994493055513, −7.12459394569051533392556285761, −6.04277838994503053468587005617, −4.87429887329860747473915124221, −3.94220621744541255463273540133, −2.26846070461465682183790784133,
2.10651816225296103656588822594, 3.99658645980633292597914057100, 4.37780362637320701844209509400, 6.42776762776722324916098787384, 7.17462390944253584190131377418, 7.79644827914627521454658374912, 9.974813691584502796203067776335, 10.75187583707726725305395624468, 11.33364166258402179209097532430, 12.83197646130156295275176999835
