L(s) = 1 | + (−0.218 − 1.39i)2-s + (0.262 − 1.71i)3-s + (−1.90 + 0.611i)4-s − 0.130·5-s + (−2.44 + 0.00827i)6-s + 3.77i·7-s + (1.27 + 2.52i)8-s + (−2.86 − 0.897i)9-s + (0.0285 + 0.182i)10-s + 4.36i·11-s + (0.547 + 3.42i)12-s + 5.50i·13-s + (5.27 − 0.826i)14-s + (−0.0342 + 0.223i)15-s + (3.25 − 2.32i)16-s − 0.402i·17-s + ⋯ |
L(s) = 1 | + (−0.154 − 0.987i)2-s + (0.151 − 0.988i)3-s + (−0.952 + 0.305i)4-s − 0.0583·5-s + (−0.999 + 0.00338i)6-s + 1.42i·7-s + (0.449 + 0.893i)8-s + (−0.954 − 0.299i)9-s + (0.00902 + 0.0576i)10-s + 1.31i·11-s + (0.158 + 0.987i)12-s + 1.52i·13-s + (1.41 − 0.220i)14-s + (−0.00883 + 0.0577i)15-s + (0.813 − 0.582i)16-s − 0.0975i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.821451 + 0.130036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.821451 + 0.130036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.218 + 1.39i)T \) |
| 3 | \( 1 + (-0.262 + 1.71i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 0.130T + 5T^{2} \) |
| 7 | \( 1 - 3.77iT - 7T^{2} \) |
| 11 | \( 1 - 4.36iT - 11T^{2} \) |
| 13 | \( 1 - 5.50iT - 13T^{2} \) |
| 17 | \( 1 + 0.402iT - 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 29 | \( 1 + 8.77T + 29T^{2} \) |
| 31 | \( 1 + 7.75iT - 31T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 + 2.80iT - 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 + 0.336T + 53T^{2} \) |
| 59 | \( 1 + 9.68iT - 59T^{2} \) |
| 61 | \( 1 - 2.78iT - 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 - 0.417T + 73T^{2} \) |
| 79 | \( 1 - 3.09iT - 79T^{2} \) |
| 83 | \( 1 + 3.59iT - 83T^{2} \) |
| 89 | \( 1 - 11.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−11.27045648211929599753347072593, −9.644835615910399603215083239681, −9.267115373428959727997763900754, −8.339696392617991072844565017926, −7.41607719906124417893626787650, −6.28031592621472733064317150973, −5.15717162721718454580734095782, −3.91855628062134273879718230733, −2.30612707626149084248724290467, −1.92536386284910978914731541491,
0.48952156571604758879591709904, 3.42258957449619005048755805274, 4.04437879173489846136653801701, 5.31067438077855226061706339202, 5.97339472638553280230884864280, 7.33452651233868926871605799340, 8.052077442578439055809508270320, 8.855076469541446940463200896622, 9.835453769119590755259199110799, 10.66199303306648061861493589653