| 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
−11.69241155305123254720763967499, −10.64819007080968829582629255294, −9.631136063472238309151452898361, −8.394718625038822548675969964468, −7.21210779407612511143300977228, −6.64860489159965531586887252917, −4.71680665237871916453024112445, −3.05428879859181998406319915029, −1.79792208749039439126191938333, −0.04956193220151610638703339253,
1.46597836021180464665815444159, 3.75517129212300838279264595417, 5.06887615717467468299665290499, 6.02134640832043806951153480160, 7.70859826536714221970008791376, 8.430460948765986340518805364748, 9.531172481718930606217589042746, 10.33727680734915160242784433006, 11.66846313167648846938672262719, 12.31632838603670277762322304159
