L(s) = 1 | + (−2.29 − 1.65i)2-s + (−5.96 − 5.96i)3-s + (2.50 + 7.59i)4-s + (8.67 − 8.67i)5-s + (3.77 + 23.5i)6-s − 1.63i·7-s + (6.86 − 21.5i)8-s + 44.1i·9-s + (−34.2 + 5.49i)10-s + (18.2 − 18.2i)11-s + (30.4 − 60.2i)12-s + (−9.34 − 9.34i)13-s + (−2.71 + 3.75i)14-s − 103.·15-s + (−51.4 + 38.0i)16-s + 53.6·17-s + ⋯ |
L(s) = 1 | + (−0.810 − 0.586i)2-s + (−1.14 − 1.14i)3-s + (0.312 + 0.949i)4-s + (0.776 − 0.776i)5-s + (0.257 + 1.60i)6-s − 0.0885i·7-s + (0.303 − 0.952i)8-s + 1.63i·9-s + (−1.08 + 0.173i)10-s + (0.498 − 0.498i)11-s + (0.731 − 1.44i)12-s + (−0.199 − 0.199i)13-s + (−0.0518 + 0.0717i)14-s − 1.78·15-s + (−0.804 + 0.594i)16-s + 0.764·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.282044 - 0.499003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.282044 - 0.499003i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.29 + 1.65i)T \) |
good | 3 | \( 1 + (5.96 + 5.96i)T + 27iT^{2} \) |
| 5 | \( 1 + (-8.67 + 8.67i)T - 125iT^{2} \) |
| 7 | \( 1 + 1.63iT - 343T^{2} \) |
| 11 | \( 1 + (-18.2 + 18.2i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (9.34 + 9.34i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-70.9 - 70.9i)T + 6.85e3iT^{2} \) |
| 23 | \( 1 - 25.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (181. + 181. i)T + 2.43e4iT^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-174. + 174. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 - 198. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (285. - 285. i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 - 78.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + (525. - 525. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (-46.5 + 46.5i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-193. - 193. i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (-282. - 282. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 727. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 106. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 58.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + (410. + 410. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 768. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 809.T + 9.12e5T^{2} \) |
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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
−18.17303671270217465371053883425, −17.12161523152205369607742367113, −16.56015012857676819898551060375, −13.49315262608994140882136299642, −12.42647682748603329334970290339, −11.40897587840056563710395050447, −9.675786585896418739096110158090, −7.74657942407784517463432430670, −5.91305368120038529267052862920, −1.22196100417709754142613348311,
5.31112394515550259827808581077, 6.74697335752028892035439759222, 9.477681158678570042125916994194, 10.32771548353112740908687031961, 11.55298207393696286132009853932, 14.34517882687107856052799216409, 15.47385542628266901040824855860, 16.71171042161424593286110844091, 17.52594130393967864454040002522, 18.55591025025149052524263731487