| L(s) = 1 | + (−3.23 + 2.35i)2-s + (−2.78 + 8.55i)3-s + (4.94 − 15.2i)4-s + (−17.5 − 53.0i)5-s + (−11.1 − 34.2i)6-s + 47.3·7-s + (19.7 + 60.8i)8-s + (−65.5 − 47.6i)9-s + (181. + 130. i)10-s + (234. − 170. i)11-s + (116. + 84.6i)12-s + (−434. − 315. i)13-s + (−153. + 111. i)14-s + (503. − 2.91i)15-s + (−207. − 150. i)16-s + (127. + 391. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.314 − 0.949i)5-s + (−0.126 − 0.388i)6-s + 0.364·7-s + (0.109 + 0.336i)8-s + (−0.269 − 0.195i)9-s + (0.574 + 0.412i)10-s + (0.583 − 0.424i)11-s + (0.233 + 0.169i)12-s + (−0.712 − 0.517i)13-s + (−0.208 + 0.151i)14-s + (0.577 − 0.00334i)15-s + (−0.202 − 0.146i)16-s + (0.106 + 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1173050186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1173050186\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (17.5 + 53.0i)T \) |
| good | 7 | \( 1 - 47.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-234. + 170. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (434. + 315. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-127. - 391. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-738. - 2.27e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.33e3 - 1.69e3i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-1.83e3 + 5.66e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.20e3 - 3.71e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.21e4 + 8.81e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (2.87e3 + 2.09e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (5.89e3 - 1.81e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-4.03e3 + 1.24e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (6.09e3 + 4.42e3i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (3.23e4 - 2.34e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-2.22e3 - 6.83e3i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.31e4 + 4.05e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (6.05e4 - 4.39e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-1.40e3 + 4.32e3i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.94e4 - 5.98e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-6.62e4 + 4.81e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (2.45e4 - 7.55e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−12.31632838603670277762322304159, −11.66846313167648846938672262719, −10.33727680734915160242784433006, −9.531172481718930606217589042746, −8.430460948765986340518805364748, −7.70859826536714221970008791376, −6.02134640832043806951153480160, −5.06887615717467468299665290499, −3.75517129212300838279264595417, −1.46597836021180464665815444159,
0.04956193220151610638703339253, 1.79792208749039439126191938333, 3.05428879859181998406319915029, 4.71680665237871916453024112445, 6.64860489159965531586887252917, 7.21210779407612511143300977228, 8.394718625038822548675969964468, 9.631136063472238309151452898361, 10.64819007080968829582629255294, 11.69241155305123254720763967499
