| L(s) = 1 | + (−0.184 + 1.16i)2-s + (1.52 + 0.827i)3-s + (0.573 + 0.186i)4-s + (−1.19 − 1.89i)5-s + (−1.24 + 1.62i)6-s + (−3.04 − 3.04i)7-s + (−1.39 + 2.74i)8-s + (1.62 + 2.51i)9-s + (2.43 − 1.04i)10-s + (−0.00182 − 0.00251i)11-s + (0.717 + 0.757i)12-s + (−1.68 + 0.266i)13-s + (4.11 − 2.98i)14-s + (−0.244 − 3.86i)15-s + (−1.96 − 1.42i)16-s + (−1.94 − 0.992i)17-s + ⋯ |
| L(s) = 1 | + (−0.130 + 0.825i)2-s + (0.878 + 0.477i)3-s + (0.286 + 0.0930i)4-s + (−0.532 − 0.846i)5-s + (−0.509 + 0.662i)6-s + (−1.14 − 1.14i)7-s + (−0.493 + 0.969i)8-s + (0.543 + 0.839i)9-s + (0.768 − 0.328i)10-s + (−0.000550 − 0.000757i)11-s + (0.207 + 0.218i)12-s + (−0.467 + 0.0740i)13-s + (1.09 − 0.798i)14-s + (−0.0631 − 0.998i)15-s + (−0.491 − 0.357i)16-s + (−0.472 − 0.240i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.908528 + 0.535116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.908528 + 0.535116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.52 - 0.827i)T \) |
| 5 | \( 1 + (1.19 + 1.89i)T \) |
| good | 2 | \( 1 + (0.184 - 1.16i)T + (-1.90 - 0.618i)T^{2} \) |
| 7 | \( 1 + (3.04 + 3.04i)T + 7iT^{2} \) |
| 11 | \( 1 + (0.00182 + 0.00251i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.68 - 0.266i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.94 + 0.992i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.48 + 0.806i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-6.89 - 1.09i)T + (21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.839 + 2.58i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.337 + 1.03i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.24 - 7.88i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (2.69 - 3.70i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.05 + 2.05i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.11 + 10.0i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (4.13 - 2.10i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (2.02 + 1.47i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.559 + 0.406i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.12 + 4.16i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-7.42 - 2.41i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.211 - 1.33i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (11.7 + 3.81i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.49 + 4.89i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.962 + 0.699i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.60 + 4.38i)T + (57.0 - 78.4i)T^{2} \) |
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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
−15.09283003469265263972638479731, −13.74267249256307648349498170983, −12.91842511786684933854448828558, −11.40233083497505723804976207371, −9.930643397671105164188924497305, −8.894914386720831175649136182971, −7.70925439091325177183825707659, −6.85502455691613276424420024744, −4.82704815878554408155176312577, −3.26912228052785336697570750381,
2.53702963825334391207199464626, 3.35661591416217176167328142296, 6.32763958488452429476323900918, 7.26999555150358466328110969260, 8.946812829929421027546837609005, 9.851666035803658439866919331117, 11.14426342241616067377596270855, 12.31523377189877753395356794288, 12.86905451733283550639703667224, 14.47628396644258773173006467347
