| L(s) = 1 | + (−1.68 + 1.24i)2-s + (0.441 − 2.33i)3-s + (0.697 − 2.26i)4-s + (−2.22 + 0.177i)5-s + (2.15 + 4.46i)6-s + (−2.04 + 1.67i)7-s + (0.253 + 0.724i)8-s + (−2.44 − 0.961i)9-s + (3.52 − 3.06i)10-s + (0.785 + 2.00i)11-s + (−4.96 − 2.62i)12-s + (0.473 + 4.20i)13-s + (1.35 − 5.36i)14-s + (−0.569 + 5.27i)15-s + (2.58 + 1.76i)16-s + (0.0695 + 1.85i)17-s + ⋯ |
| L(s) = 1 | + (−1.18 + 0.877i)2-s + (0.254 − 1.34i)3-s + (0.348 − 1.13i)4-s + (−0.996 + 0.0794i)5-s + (0.878 + 1.82i)6-s + (−0.773 + 0.634i)7-s + (0.0896 + 0.256i)8-s + (−0.816 − 0.320i)9-s + (1.11 − 0.969i)10-s + (0.236 + 0.603i)11-s + (−1.43 − 0.757i)12-s + (0.131 + 1.16i)13-s + (0.362 − 1.43i)14-s + (−0.146 + 1.36i)15-s + (0.646 + 0.440i)16-s + (0.0168 + 0.450i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.147910 + 0.288071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147910 + 0.288071i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.22 - 0.177i)T \) |
| 7 | \( 1 + (2.04 - 1.67i)T \) |
| good | 2 | \( 1 + (1.68 - 1.24i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.441 + 2.33i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.785 - 2.00i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.473 - 4.20i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.0695 - 1.85i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-3.30 - 5.72i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (7.01 + 0.262i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (4.54 + 1.03i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (3.46 + 1.99i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.180 - 0.342i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (4.04 - 8.40i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (8.58 + 3.00i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-3.02 - 4.09i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-0.0786 - 0.148i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.00101 + 0.0136i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-3.58 - 11.6i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (1.30 + 4.88i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (1.72 + 7.57i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.35 - 5.89i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-5.78 + 3.34i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.9 - 1.34i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-0.169 + 0.432i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \) |
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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
−12.22176435783865864957522347505, −11.82176498322654885183141089675, −10.11181160620898396993394230740, −9.191708623813706000271037637978, −8.212043861653109963221178003459, −7.60678457973631133377838833230, −6.77074193408872388120975169252, −6.04875775532680259083819933765, −3.75542197199676792880545010636, −1.73759264536202186722820603256,
0.37057614115912085009513821775, 3.14327855798401255247348123823, 3.72771596438222983083672829257, 5.25321700295534288247390865885, 7.22262382467630691382968834107, 8.287577823055967851443924362932, 9.137339562508609368089634400530, 9.890555545102660197987773263711, 10.64841468638037927683658698431, 11.26875149233948804508258642205
