L(s) = 1 | + (0.781 + 0.623i)2-s + (0.974 + 0.222i)3-s + (0.222 + 0.974i)4-s + (0.623 + 0.781i)6-s + (−0.974 − 0.222i)7-s + (−0.433 + 0.900i)8-s + (0.900 + 0.433i)9-s + (−0.900 + 0.433i)11-s + i·12-s + (0.433 + 0.900i)13-s + (−0.623 − 0.781i)14-s + (−0.900 + 0.433i)16-s + i·17-s + (0.433 + 0.900i)18-s + (0.222 + 0.974i)19-s + ⋯ |
L(s) = 1 | + (0.781 + 0.623i)2-s + (0.974 + 0.222i)3-s + (0.222 + 0.974i)4-s + (0.623 + 0.781i)6-s + (−0.974 − 0.222i)7-s + (−0.433 + 0.900i)8-s + (0.900 + 0.433i)9-s + (−0.900 + 0.433i)11-s + i·12-s + (0.433 + 0.900i)13-s + (−0.623 − 0.781i)14-s + (−0.900 + 0.433i)16-s + i·17-s + (0.433 + 0.900i)18-s + (0.222 + 0.974i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.390104651 + 2.920988530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390104651 + 2.920988530i\) |
\(L(1)\) |
\(\approx\) |
\(1.563504538 + 1.207390012i\) |
\(L(1)\) |
\(\approx\) |
\(1.563504538 + 1.207390012i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.781 + 0.623i)T \) |
| 3 | \( 1 + (0.974 + 0.222i)T \) |
| 7 | \( 1 + (-0.974 - 0.222i)T \) |
| 11 | \( 1 + (-0.900 + 0.433i)T \) |
| 13 | \( 1 + (0.433 + 0.900i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.222 + 0.974i)T \) |
| 23 | \( 1 + (0.781 - 0.623i)T \) |
| 31 | \( 1 + (0.623 - 0.781i)T \) |
| 37 | \( 1 + (0.433 - 0.900i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.781 - 0.623i)T \) |
| 47 | \( 1 + (-0.433 - 0.900i)T \) |
| 53 | \( 1 + (-0.781 - 0.623i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.222 + 0.974i)T \) |
| 67 | \( 1 + (0.433 - 0.900i)T \) |
| 71 | \( 1 + (-0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.781 - 0.623i)T \) |
| 79 | \( 1 + (0.900 + 0.433i)T \) |
| 83 | \( 1 + (-0.974 + 0.222i)T \) |
| 89 | \( 1 + (-0.623 + 0.781i)T \) |
| 97 | \( 1 + (0.974 - 0.222i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.78142330307608470051768460620, −26.5734981985901831374501529668, −25.48328944198225294939716532670, −24.71233555087241619160427474709, −23.581376778992677467751721765314, −22.66463746615652521696105745747, −21.5362088787779912056973429306, −20.66859684179690139989642610108, −19.78489511055690594150574404517, −18.97645043823874803926459176758, −18.07521223728700887569157234608, −15.80813839149745746049604077055, −15.51931174032599236211511321139, −14.05639241893355216554413692993, −13.23007526262462079022553292073, −12.65138301542592404806713572537, −11.14654516075862890034818613033, −9.93574255780786871555317781644, −9.02428344703781495702468281839, −7.478082903246491166312042093048, −6.191739076339106929574817327230, −4.82655209326657088060190948586, −3.118423413045211067614933568275, −2.82169118505347853862116801097, −0.858760484549227729981093289150,
2.27834028693999448066600244983, 3.51167958618677170964193424229, 4.42482201713998204792157248748, 6.0294193808627128317223042115, 7.197335984769336204475584589234, 8.21671805633610972719891972865, 9.39167630516568135806546517670, 10.66499786717061627155224907470, 12.474855678787409019006040548031, 13.16663350895456927297512578367, 14.1270221462478257166545092277, 15.121939460219900850798561618633, 16.00022294351709568365042288216, 16.80922248338889549845213282417, 18.44878639742276980029799379167, 19.496809949700664435992776464640, 20.73312715266629882720186406218, 21.29535657774222645267805467036, 22.53825751992188112612570175911, 23.42716263351227462754617684925, 24.475938447373760578832711065089, 25.522367491553697013878342725247, 26.14449260554272707396919111720, 26.77020336045224644637192697236, 28.45793849221921929150130439432