| L(s) = 1 | + (−3.23 − 2.35i)2-s + (−2.78 − 8.55i)3-s + (4.94 + 15.2i)4-s + (−54.7 − 11.1i)5-s + (−11.1 + 34.2i)6-s − 236.·7-s + (19.7 − 60.8i)8-s + (−65.5 + 47.6i)9-s + (150. + 164. i)10-s + (−550. − 399. i)11-s + (116. − 84.6i)12-s + (392. − 285. i)13-s + (765. + 556. i)14-s + (56.7 + 499. i)15-s + (−207. + 150. i)16-s + (−573. + 1.76e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.979 − 0.199i)5-s + (−0.126 + 0.388i)6-s − 1.82·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.477 + 0.521i)10-s + (−1.37 − 0.996i)11-s + (0.233 − 0.169i)12-s + (0.644 − 0.468i)13-s + (1.04 + 0.758i)14-s + (0.0650 + 0.573i)15-s + (−0.202 + 0.146i)16-s + (−0.481 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1971126161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1971126161\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 3 | \( 1 + (2.78 + 8.55i)T \) |
| 5 | \( 1 + (54.7 + 11.1i)T \) |
| good | 7 | \( 1 + 236.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (550. + 399. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-392. + 285. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (573. - 1.76e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-468. + 1.44e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.38e3 + 1.72e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (115. + 355. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (367. - 1.12e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.10e4 - 8.04e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.22e4 + 8.89e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (5.46e3 + 1.68e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (38.6 + 118. i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-6.35e3 + 4.61e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-9.64e3 - 7.00e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (6.89e3 - 2.12e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (2.13e4 + 6.55e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (7.56e3 + 5.49e3i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-5.53e3 - 1.70e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (3.44e4 - 1.06e5i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (8.06e4 + 5.86e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (2.05e4 + 6.31e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−12.25718408733962384416531069188, −10.93983972494525435139814133565, −10.34330559946983064015069141750, −8.812934166804744365837625216870, −8.121212900242104330523509612531, −6.89997481015182620405743422553, −5.82631872226970501860599685424, −3.74474627672886110080167001710, −2.76102226713123497706740012579, −0.55356113827875513171446822707,
0.16349987018459745848337215372, 2.83317197792116465933436986919, 4.14253979784251520546668042760, 5.67445973131160463601600653228, 6.91971569023168566161526481686, 7.74227376345000715139525579713, 9.208829080151399226261514122722, 9.896578671107734655103053677127, 10.82355989768238635042566438373, 12.00769426442480656013931027362
