L(s) = 1 | + (0.382 + 0.662i)2-s + (0.258 − 0.965i)3-s + (0.207 − 0.358i)4-s + (−0.923 + 0.382i)5-s + (0.739 − 0.198i)6-s + 1.08·8-s + (−0.866 − 0.499i)9-s + (−0.607 − 0.465i)10-s + (1.78 + 0.478i)11-s + (−0.292 − 0.292i)12-s + (0.130 + 0.991i)15-s + (0.207 + 0.358i)16-s − 0.765i·18-s + (−0.0540 + 0.410i)20-s + (0.366 + 1.36i)22-s + ⋯ |
L(s) = 1 | + (0.382 + 0.662i)2-s + (0.258 − 0.965i)3-s + (0.207 − 0.358i)4-s + (−0.923 + 0.382i)5-s + (0.739 − 0.198i)6-s + 1.08·8-s + (−0.866 − 0.499i)9-s + (−0.607 − 0.465i)10-s + (1.78 + 0.478i)11-s + (−0.292 − 0.292i)12-s + (0.130 + 0.991i)15-s + (0.207 + 0.358i)16-s − 0.765i·18-s + (−0.0540 + 0.410i)20-s + (0.366 + 1.36i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.678176101\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678176101\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (0.923 - 0.382i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.198 - 0.739i)T + (-0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + 1.84iT - T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 - 1.84iT - T^{2} \) |
| 89 | \( 1 + (0.478 - 1.78i)T + (-0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.793774455633894134709991111915, −8.030277011390276904973924627765, −7.25070689967832553540468594586, −6.77011864929914513328852821513, −6.32476613629273202971014779333, −5.29333721014773647476003712318, −4.22534639483850601836248957113, −3.51318226968154996938548021224, −2.18362194820203819793485076392, −1.15553365863738361731840772855,
1.42302235949739191032877535131, 2.87139963909498344998926182180, 3.56646495489308231269044628064, 4.18375149080599939380770025952, 4.69413675210090257473663368399, 5.89197988900798868631341168736, 6.92814071894990277507651667084, 7.77345701440304190430542800854, 8.556849487178646622096516308060, 9.061345325806435322276805688584