| L(s) = 1 | + (0.796 − 1.07i)2-s + (2.57 − 0.487i)3-s + (0.0592 + 0.191i)4-s + (−1.63 − 1.52i)5-s + (1.52 − 3.16i)6-s + (−1.87 + 1.87i)7-s + (2.78 + 0.975i)8-s + (3.60 − 1.41i)9-s + (−2.94 + 0.550i)10-s + (1.32 − 3.37i)11-s + (0.246 + 0.465i)12-s + (−3.93 − 0.443i)13-s + (0.528 + 3.50i)14-s + (−4.95 − 3.13i)15-s + (2.93 − 2.00i)16-s + (3.16 + 0.118i)17-s + ⋯ |
| L(s) = 1 | + (0.563 − 0.763i)2-s + (1.48 − 0.281i)3-s + (0.0296 + 0.0959i)4-s + (−0.731 − 0.681i)5-s + (0.623 − 1.29i)6-s + (−0.707 + 0.707i)7-s + (0.985 + 0.344i)8-s + (1.20 − 0.472i)9-s + (−0.932 + 0.174i)10-s + (0.399 − 1.01i)11-s + (0.0710 + 0.134i)12-s + (−1.09 − 0.122i)13-s + (0.141 + 0.937i)14-s + (−1.28 − 0.808i)15-s + (0.734 − 0.501i)16-s + (0.767 + 0.0287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.95907 - 1.10377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95907 - 1.10377i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.63 + 1.52i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
| good | 2 | \( 1 + (-0.796 + 1.07i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (-2.57 + 0.487i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.32 + 3.37i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (3.93 + 0.443i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-3.16 - 0.118i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (3.27 - 5.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0875 - 2.33i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (8.14 - 1.85i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.63 - 0.943i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.33 + 4.40i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-0.959 - 1.99i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (3.38 + 9.67i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (5.98 + 4.41i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-7.35 - 3.88i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.252 + 3.37i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.195 - 0.634i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-8.69 - 2.32i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.32 + 5.78i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-4.08 + 3.01i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-2.13 - 1.22i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.677 + 6.01i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-0.102 - 0.261i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (5.06 + 5.06i)T + 97iT^{2} \) |
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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
−12.29839726401895129386331807748, −11.33903593603007438089089862762, −9.870794549237381201363298716451, −8.910716150416257191970780549840, −8.116787245485526268007924321264, −7.37352600759729600703351613385, −5.50707124739945442915626876968, −3.83183908589599364484026600701, −3.31754748795019057423942246424, −2.02599570248996344167320876266,
2.53164986526252305243098415166, 3.86130021148521521845048229035, 4.62469401683225376943723811266, 6.57341592909797654282647913336, 7.28206921464224121637673493711, 7.923906144389432192240835957963, 9.530157119423717600791245594849, 9.971606864177265654500293966922, 11.18735422194577275584810536310, 12.70823698972980920550797551276
