L(s) = 1 | + (−0.866 − 0.5i)4-s + (1.67 + 0.448i)7-s + (0.499 + 0.866i)16-s + i·19-s + (−1.22 − 1.22i)28-s + (−0.5 + 0.866i)31-s + (1.22 − 1.22i)37-s + (0.448 − 1.67i)43-s + (1.73 + 1.00i)49-s + (0.5 + 0.866i)61-s − 0.999i·64-s + (1.22 + 1.22i)73-s + (0.5 − 0.866i)76-s + (−0.866 + 0.5i)79-s + (−1.67 − 0.448i)97-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)4-s + (1.67 + 0.448i)7-s + (0.499 + 0.866i)16-s + i·19-s + (−1.22 − 1.22i)28-s + (−0.5 + 0.866i)31-s + (1.22 − 1.22i)37-s + (0.448 − 1.67i)43-s + (1.73 + 1.00i)49-s + (0.5 + 0.866i)61-s − 0.999i·64-s + (1.22 + 1.22i)73-s + (0.5 − 0.866i)76-s + (−0.866 + 0.5i)79-s + (−1.67 − 0.448i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.152325098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152325098\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 79 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.199704261167514198116804658283, −8.552115691129193745667829730372, −8.018211899221571245435642590047, −7.13146079529074366842506003617, −5.75622041814312355826131836706, −5.43212924191603903644045174762, −4.51692093088414541889381651128, −3.81726409829568152142936679061, −2.23017344388341383376505677678, −1.26734063687963245963115861840,
1.09699256581154718568168489742, 2.47583106525465690559135112607, 3.70377690339327215326203808305, 4.68316913927023268363890228736, 4.88015897349262687325246178872, 6.09390801263172816929505254309, 7.30207478789387664035889052363, 7.88616168199569739273788478554, 8.414017307325871895772966280758, 9.250268966268234403334913311055