L(s) = 1 | − 28.3i·2-s + 796.·3-s + 3.29e3·4-s + 5.72e3·5-s − 2.26e4i·6-s + 2.02e5i·7-s − 2.09e5i·8-s + 1.03e5·9-s − 1.62e5i·10-s + (1.72e6 + 4.24e5i)11-s + 2.62e6·12-s − 6.56e6i·13-s + 5.74e6·14-s + 4.55e6·15-s + 7.53e6·16-s + 2.59e7i·17-s + ⋯ |
L(s) = 1 | − 0.443i·2-s + 1.09·3-s + 0.803·4-s + 0.366·5-s − 0.484i·6-s + 1.72i·7-s − 0.799i·8-s + 0.195·9-s − 0.162i·10-s + (0.970 + 0.239i)11-s + 0.878·12-s − 1.36i·13-s + 0.763·14-s + 0.400·15-s + 0.448·16-s + 1.07i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(3.02093 - 0.367053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.02093 - 0.367053i\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-1.72e6 - 4.24e5i)T \) |
good | 2 | \( 1 + 28.3iT - 4.09e3T^{2} \) |
| 3 | \( 1 - 796.T + 5.31e5T^{2} \) |
| 5 | \( 1 - 5.72e3T + 2.44e8T^{2} \) |
| 7 | \( 1 - 2.02e5iT - 1.38e10T^{2} \) |
| 13 | \( 1 + 6.56e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 - 2.59e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 7.15e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 9.33e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + 6.46e7iT - 3.53e17T^{2} \) |
| 31 | \( 1 - 5.07e8T + 7.87e17T^{2} \) |
| 37 | \( 1 - 5.29e8T + 6.58e18T^{2} \) |
| 41 | \( 1 - 4.97e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 1.97e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + 1.74e10T + 1.16e20T^{2} \) |
| 53 | \( 1 + 1.99e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 2.01e10T + 1.77e21T^{2} \) |
| 61 | \( 1 - 5.39e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 - 7.88e10T + 8.18e21T^{2} \) |
| 71 | \( 1 - 4.09e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + 2.05e10iT - 2.29e22T^{2} \) |
| 79 | \( 1 + 9.65e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + 2.46e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 1.49e11T + 2.46e23T^{2} \) |
| 97 | \( 1 - 5.12e11T + 6.93e23T^{2} \) |
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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
−17.66762879692539700004732699198, −15.52177107227388784100130885295, −14.85571342406667155643891900682, −12.93473329103268348797840565814, −11.60739584068539514869174042177, −9.648463989246751415028494179167, −8.292872737519272689304021340976, −6.06503730892599228626762514641, −3.05489240101678304621905459590, −1.98853782769296521458150469022,
1.73080062762996615624344506108, 3.75394629712621173613568732013, 6.60415488300852548599730045570, 7.896147381218841504840845916474, 9.735823892903834970446721576369, 11.51865714939836694291215432498, 14.05729196748031930650246394085, 14.20198259297611738686251916246, 16.26032134773203303511638247806, 17.12832332032380435712712624494