| L(s) = 1 | + (−7.37 + 8.51i)2-s + (−13.1 + 8.42i)3-s + (−8.96 − 62.3i)4-s + (186. − 85.2i)5-s + (24.9 − 173. i)6-s + (65.3 + 222. i)7-s + (−9.86 − 6.33i)8-s + (100. − 221. i)9-s + (−651. + 2.21e3i)10-s + (−1.47e3 + 1.27e3i)11-s + (642. + 741. i)12-s + (−1.70e3 − 500. i)13-s + (−2.37e3 − 1.08e3i)14-s + (−1.72e3 + 2.69e3i)15-s + (3.99e3 − 1.17e3i)16-s + (−235. − 33.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.922 + 1.06i)2-s + (−0.485 + 0.312i)3-s + (−0.140 − 0.973i)4-s + (1.49 − 0.681i)5-s + (0.115 − 0.804i)6-s + (0.190 + 0.649i)7-s + (−0.0192 − 0.0123i)8-s + (0.138 − 0.303i)9-s + (−0.651 + 2.21i)10-s + (−1.10 + 0.957i)11-s + (0.371 + 0.429i)12-s + (−0.775 − 0.227i)13-s + (−0.866 − 0.395i)14-s + (−0.512 + 0.797i)15-s + (0.974 − 0.286i)16-s + (−0.0479 − 0.00689i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0922329 - 0.129561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0922329 - 0.129561i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (13.1 - 8.42i)T \) |
| 23 | \( 1 + (5.30e3 - 1.09e4i)T \) |
| good | 2 | \( 1 + (7.37 - 8.51i)T + (-9.10 - 63.3i)T^{2} \) |
| 5 | \( 1 + (-186. + 85.2i)T + (1.02e4 - 1.18e4i)T^{2} \) |
| 7 | \( 1 + (-65.3 - 222. i)T + (-9.89e4 + 6.36e4i)T^{2} \) |
| 11 | \( 1 + (1.47e3 - 1.27e3i)T + (2.52e5 - 1.75e6i)T^{2} \) |
| 13 | \( 1 + (1.70e3 + 500. i)T + (4.06e6 + 2.60e6i)T^{2} \) |
| 17 | \( 1 + (235. + 33.8i)T + (2.31e7 + 6.80e6i)T^{2} \) |
| 19 | \( 1 + (3.80e3 - 546. i)T + (4.51e7 - 1.32e7i)T^{2} \) |
| 29 | \( 1 + (-496. + 3.45e3i)T + (-5.70e8 - 1.67e8i)T^{2} \) |
| 31 | \( 1 + (2.70e4 + 1.74e4i)T + (3.68e8 + 8.07e8i)T^{2} \) |
| 37 | \( 1 + (5.95e4 + 2.71e4i)T + (1.68e9 + 1.93e9i)T^{2} \) |
| 41 | \( 1 + (-585. - 1.28e3i)T + (-3.11e9 + 3.58e9i)T^{2} \) |
| 43 | \( 1 + (5.39e4 + 8.39e4i)T + (-2.62e9 + 5.75e9i)T^{2} \) |
| 47 | \( 1 + 1.31e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + (-2.62e4 - 8.93e4i)T + (-1.86e10 + 1.19e10i)T^{2} \) |
| 59 | \( 1 + (-8.01e4 - 2.35e4i)T + (3.54e10 + 2.28e10i)T^{2} \) |
| 61 | \( 1 + (-3.09e4 + 4.81e4i)T + (-2.14e10 - 4.68e10i)T^{2} \) |
| 67 | \( 1 + (-3.29e5 - 2.85e5i)T + (1.28e10 + 8.95e10i)T^{2} \) |
| 71 | \( 1 + (4.47e5 - 5.16e5i)T + (-1.82e10 - 1.26e11i)T^{2} \) |
| 73 | \( 1 + (5.84e4 + 4.06e5i)T + (-1.45e11 + 4.26e10i)T^{2} \) |
| 79 | \( 1 + (-2.19e5 + 7.49e5i)T + (-2.04e11 - 1.31e11i)T^{2} \) |
| 83 | \( 1 + (6.52e5 + 2.97e5i)T + (2.14e11 + 2.47e11i)T^{2} \) |
| 89 | \( 1 + (8.47e4 + 1.31e5i)T + (-2.06e11 + 4.52e11i)T^{2} \) |
| 97 | \( 1 + (9.95e5 - 4.54e5i)T + (5.45e11 - 6.29e11i)T^{2} \) |
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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
−14.71947261565289497459910763502, −13.15551324756835817882968538604, −12.23005825344478496695562639743, −10.24106862476072497051560997219, −9.643340904250311896416088729421, −8.621388844846078059047115879045, −7.23376989079327184398856275426, −5.77896085624102806234777679266, −5.16369627926658322776361354270, −2.00996536733274328656560467161,
0.083480313015793408547556883694, 1.69732734900914862379907351650, 2.81246900947390333510590439990, 5.40356114965424661177562276139, 6.70654017954890912260204118687, 8.317423145225347197231242169641, 9.778066578154024657063249059452, 10.50743764659744269005510484833, 11.10380725019989950969494182559, 12.59078334925095157671957439146
