L(s) = 1 | + (−0.239 + 0.970i)4-s + (0.819 + 1.82i)7-s + i·13-s + (−0.885 − 0.464i)16-s + (−1.11 − 1.11i)19-s + (−0.822 − 0.568i)25-s + (−1.96 + 0.359i)28-s + (0.186 − 1.01i)31-s + (0.807 + 0.147i)37-s + (1.53 + 1.06i)43-s + (−1.97 + 2.23i)49-s + (−0.970 − 0.239i)52-s + (0.169 + 0.447i)61-s + (0.663 − 0.748i)64-s + (1.68 + 1.01i)67-s + ⋯ |
L(s) = 1 | + (−0.239 + 0.970i)4-s + (0.819 + 1.82i)7-s + i·13-s + (−0.885 − 0.464i)16-s + (−1.11 − 1.11i)19-s + (−0.822 − 0.568i)25-s + (−1.96 + 0.359i)28-s + (0.186 − 1.01i)31-s + (0.807 + 0.147i)37-s + (1.53 + 1.06i)43-s + (−1.97 + 2.23i)49-s + (−0.970 − 0.239i)52-s + (0.169 + 0.447i)61-s + (0.663 − 0.748i)64-s + (1.68 + 1.01i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.036485767\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.036485767\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + (0.239 - 0.970i)T^{2} \) |
| 5 | \( 1 + (0.822 + 0.568i)T^{2} \) |
| 7 | \( 1 + (-0.819 - 1.82i)T + (-0.663 + 0.748i)T^{2} \) |
| 11 | \( 1 + (0.239 + 0.970i)T^{2} \) |
| 17 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 19 | \( 1 + (1.11 + 1.11i)T + iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.970 - 0.239i)T^{2} \) |
| 31 | \( 1 + (-0.186 + 1.01i)T + (-0.935 - 0.354i)T^{2} \) |
| 37 | \( 1 + (-0.807 - 0.147i)T + (0.935 + 0.354i)T^{2} \) |
| 41 | \( 1 + (0.992 + 0.120i)T^{2} \) |
| 43 | \( 1 + (-1.53 - 1.06i)T + (0.354 + 0.935i)T^{2} \) |
| 47 | \( 1 + (-0.464 - 0.885i)T^{2} \) |
| 53 | \( 1 + (-0.748 - 0.663i)T^{2} \) |
| 59 | \( 1 + (-0.822 - 0.568i)T^{2} \) |
| 61 | \( 1 + (-0.169 - 0.447i)T + (-0.748 + 0.663i)T^{2} \) |
| 67 | \( 1 + (-1.68 - 1.01i)T + (0.464 + 0.885i)T^{2} \) |
| 71 | \( 1 + (0.992 + 0.120i)T^{2} \) |
| 73 | \( 1 + (1.50 + 1.17i)T + (0.239 + 0.970i)T^{2} \) |
| 79 | \( 1 + (-1.28 + 0.317i)T + (0.885 - 0.464i)T^{2} \) |
| 83 | \( 1 + (-0.992 + 0.120i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 + (-0.420 - 1.35i)T + (-0.822 + 0.568i)T^{2} \) |
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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
−9.416571833774114790261859214071, −9.107835818937661724347602288434, −8.312121936902710123535124743710, −7.78155106644209110986011236776, −6.61413777004465610796918176663, −5.84118785994550063574821194985, −4.72735768435643746744280105515, −4.16820956018082665465489420054, −2.65248830099296009047559570921, −2.16402069706388885519862439355,
0.862742069448653384317684905538, 1.92361089305999699726085057893, 3.68006184048888900606337614167, 4.36334491589433122769255142002, 5.25278452429318894184277176326, 6.09602418645365132031122872382, 7.06745658456971536662610556955, 7.82360789286027383820261564758, 8.515163461678512988620272377845, 9.689024552448337990382028341534