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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3251040997\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3251040997\) |
\(L(1)\) |
|
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
−9.007670053843143230021880248472, −8.677122435399653111485034663596, −7.45313288667385768139259631229, −6.85417416151347117487740303300, −5.86938762926907327127788079498, −5.13752030990921672443298771811, −3.98409690292216949519690013002, −3.35954702445305946481685309888, −2.41064502186144971767813012510, −0.24698312243357403902133226354,
1.37784500528539849099240442991, 3.07988077389639640767835194202, 3.73714602272312151146202088163, 4.67757467044621581017873021551, 5.67126400352412529284756474117, 6.34877200950377319587669253144, 7.07579040120720212380227924452, 8.144494810368734425150296267648, 8.929179771742752671953431014663, 9.645519510331246066137152972092