| L(s) = 1 | + (−2.11 + 1.56i)2-s + (0.536 − 2.83i)3-s + (1.45 − 4.72i)4-s + (1.93 + 1.11i)5-s + (3.29 + 6.85i)6-s + (1.30 − 2.30i)7-s + (2.55 + 7.31i)8-s + (−4.96 − 1.94i)9-s + (−5.85 + 0.674i)10-s + (1.23 + 3.14i)11-s + (−12.6 − 6.66i)12-s + (−0.330 − 2.93i)13-s + (0.843 + 6.91i)14-s + (4.19 − 4.90i)15-s + (−8.69 − 5.92i)16-s + (−0.170 − 4.54i)17-s + ⋯ |
| L(s) = 1 | + (−1.49 + 1.10i)2-s + (0.309 − 1.63i)3-s + (0.728 − 2.36i)4-s + (0.867 + 0.497i)5-s + (1.34 + 2.79i)6-s + (0.492 − 0.870i)7-s + (0.904 + 2.58i)8-s + (−1.65 − 0.649i)9-s + (−1.85 + 0.213i)10-s + (0.372 + 0.949i)11-s + (−3.63 − 1.92i)12-s + (−0.0915 − 0.812i)13-s + (0.225 + 1.84i)14-s + (1.08 − 1.26i)15-s + (−2.17 − 1.48i)16-s + (−0.0412 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.720525 - 0.309232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.720525 - 0.309232i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.93 - 1.11i)T \) |
| 7 | \( 1 + (-1.30 + 2.30i)T \) |
| good | 2 | \( 1 + (2.11 - 1.56i)T + (0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-0.536 + 2.83i)T + (-2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.23 - 3.14i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (0.330 + 2.93i)T + (-12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.170 + 4.54i)T + (-16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-0.821 - 1.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.84 + 0.0691i)T + (22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (4.06 + 0.927i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.56 - 2.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.543 - 1.02i)T + (-20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-5.06 + 10.5i)T + (-25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (3.82 + 1.33i)T + (33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-0.617 - 0.836i)T + (-13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-1.93 - 3.65i)T + (-29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.814 - 10.8i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (0.253 + 0.823i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.810 + 3.02i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.98 - 13.0i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.63 - 3.57i)T + (-21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (0.541 - 0.312i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.80 + 0.315i)T + (80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (2.31 - 5.90i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + (-8.62 - 8.62i)T + 97iT^{2} \) |
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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
−11.85527064350840746407943816345, −10.61403525866829578703581702915, −9.818711208467529460629499772617, −8.793621093886751334362482517681, −7.62005419703921956926044376770, −7.30429897265654985764853910522, −6.52309611554787752774949016112, −5.45196939309743877904998676844, −2.23084034846434195088778559190, −1.07803984544684849172706421733,
1.90959438913470284437184292535, 3.23183815267550936349036493875, 4.53360287001839194127530805891, 6.02558145677655737690215907144, 8.225134547701723504654910903727, 8.787282630058081765310603650543, 9.422381375342854832845738161561, 10.03785762269803634305821076300, 11.06082778792861075455820031614, 11.57795109919899492795437002874
