L(s) = 1 | + (−1.39 + 0.234i)2-s + 1.37i·3-s + (1.89 − 0.653i)4-s + i·5-s + (−0.323 − 1.92i)6-s − 2.31·7-s + (−2.48 + 1.35i)8-s + 1.09·9-s + (−0.234 − 1.39i)10-s − 1.65·11-s + (0.902 + 2.60i)12-s − 3.48·13-s + (3.23 − 0.543i)14-s − 1.37·15-s + (3.14 − 2.47i)16-s + 5.56i·17-s + ⋯ |
L(s) = 1 | + (−0.986 + 0.165i)2-s + 0.796i·3-s + (0.945 − 0.326i)4-s + 0.447i·5-s + (−0.132 − 0.785i)6-s − 0.876·7-s + (−0.877 + 0.478i)8-s + 0.365·9-s + (−0.0741 − 0.441i)10-s − 0.498·11-s + (0.260 + 0.752i)12-s − 0.965·13-s + (0.864 − 0.145i)14-s − 0.356·15-s + (0.786 − 0.617i)16-s + 1.35i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0314563 - 0.342703i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0314563 - 0.342703i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.39 - 0.234i)T \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-0.716 + 4.74i)T \) |
good | 3 | \( 1 - 1.37iT - 3T^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 - 5.56iT - 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 3.53iT - 31T^{2} \) |
| 37 | \( 1 + 9.38iT - 37T^{2} \) |
| 41 | \( 1 + 6.37T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 8.60iT - 47T^{2} \) |
| 53 | \( 1 + 2.04iT - 53T^{2} \) |
| 59 | \( 1 + 9.50iT - 59T^{2} \) |
| 61 | \( 1 - 3.86iT - 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 - 7.48iT - 89T^{2} \) |
| 97 | \( 1 - 19.0iT - 97T^{2} \) |
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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
−10.97905720465839030031123795779, −10.30076547988111313415742705115, −10.01790193540168574488168179489, −8.934590607633305572169120491207, −8.081750750435795361008093757841, −6.87139846378014611642367106366, −6.29618431420297689951516176713, −4.86348687759253197274295865945, −3.49937282298854765349123373412, −2.21987329288816431430118782926,
0.26896038116575046714917981805, 1.92165193776997459479685864167, 3.09716006771431239287285727451, 4.85065337609703566770036205748, 6.36849086160782958972175473351, 7.01216298812188685192948867182, 7.84604143654277777723465532083, 8.756459944852022528434182342396, 9.834187679370817247974272051979, 10.18736175377067345387576407784