| L(s) = 1 | + (−0.831 − 1.14i)2-s + (2.78 + 1.10i)3-s + (−0.618 + 1.90i)4-s + (−4.36 + 2.44i)5-s + (−1.05 − 4.11i)6-s + 4.66·7-s + (2.68 − 0.874i)8-s + (6.55 + 6.16i)9-s + (6.42 + 2.96i)10-s + (2.09 + 2.88i)11-s + (−3.82 + 4.62i)12-s + (18.2 + 13.2i)13-s + (−3.87 − 5.33i)14-s + (−14.8 + 1.98i)15-s + (−3.23 − 2.35i)16-s + (−11.5 + 3.76i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 − 0.572i)2-s + (0.929 + 0.368i)3-s + (−0.154 + 0.475i)4-s + (−0.872 + 0.488i)5-s + (−0.175 − 0.685i)6-s + 0.666·7-s + (0.336 − 0.109i)8-s + (0.728 + 0.685i)9-s + (0.642 + 0.296i)10-s + (0.190 + 0.261i)11-s + (−0.318 + 0.385i)12-s + (1.40 + 1.01i)13-s + (−0.276 − 0.381i)14-s + (−0.991 + 0.132i)15-s + (−0.202 − 0.146i)16-s + (−0.681 + 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44108 + 0.333103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44108 + 0.333103i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.831 + 1.14i)T \) |
| 3 | \( 1 + (-2.78 - 1.10i)T \) |
| 5 | \( 1 + (4.36 - 2.44i)T \) |
| good | 7 | \( 1 - 4.66T + 49T^{2} \) |
| 11 | \( 1 + (-2.09 - 2.88i)T + (-37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-18.2 - 13.2i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (11.5 - 3.76i)T + (233. - 169. i)T^{2} \) |
| 19 | \( 1 + (-3.82 - 11.7i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + (7.54 + 10.3i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-26.8 - 8.71i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (16.6 + 51.1i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (12.3 + 8.96i)T + (423. + 1.30e3i)T^{2} \) |
| 41 | \( 1 + (-14.9 + 20.6i)T + (-519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + 31.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (18.9 + 6.15i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (82.7 + 26.8i)T + (2.27e3 + 1.65e3i)T^{2} \) |
| 59 | \( 1 + (-34.5 + 47.6i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-59.7 + 43.4i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-9.87 - 30.3i)T + (-3.63e3 + 2.63e3i)T^{2} \) |
| 71 | \( 1 + (-106. - 34.7i)T + (4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (109. - 79.5i)T + (1.64e3 - 5.06e3i)T^{2} \) |
| 79 | \( 1 + (-4.15 + 12.7i)T + (-5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-83.2 + 27.0i)T + (5.57e3 - 4.04e3i)T^{2} \) |
| 89 | \( 1 + (-54.4 - 74.9i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-20.7 + 63.8i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−12.80307130918144475493224982652, −11.50044298698269298613815457429, −10.94741567510912951385779028696, −9.773763176439468976915966806994, −8.595901647615823604446404542394, −8.043033572471517092023705608292, −6.75197664704661889817268183278, −4.40169949426884039064509632778, −3.60082853422391343992331197365, −1.93266632142133069204948306750,
1.16680641128113921784352879925, 3.42217835664291813790249420301, 4.86819885641500922697352212670, 6.54002101750821747673596868384, 7.76723563609833541817834174233, 8.404926369272112249285341984793, 9.092349161892580165366475155232, 10.63629249969477733800933414058, 11.68792630233694654512303903387, 12.96075186872440264667459107782
