| L(s) = 1 | + (−0.685 − 0.349i)2-s + (1.03 − 1.38i)3-s + (−0.827 − 1.13i)4-s + (−2.01 − 0.962i)5-s + (−1.19 + 0.587i)6-s + (1.53 + 1.53i)7-s + (0.410 + 2.58i)8-s + (−0.843 − 2.87i)9-s + (1.04 + 1.36i)10-s + (4.90 − 1.59i)11-s + (−2.43 − 0.0357i)12-s + (1.29 + 2.54i)13-s + (−0.517 − 1.59i)14-s + (−3.43 + 1.79i)15-s + (−0.246 + 0.759i)16-s + (−0.429 + 0.0680i)17-s + ⋯ |
| L(s) = 1 | + (−0.484 − 0.246i)2-s + (0.599 − 0.800i)3-s + (−0.413 − 0.569i)4-s + (−0.902 − 0.430i)5-s + (−0.488 + 0.239i)6-s + (0.581 + 0.581i)7-s + (0.145 + 0.915i)8-s + (−0.281 − 0.959i)9-s + (0.331 + 0.431i)10-s + (1.47 − 0.480i)11-s + (−0.703 − 0.0103i)12-s + (0.360 + 0.706i)13-s + (−0.138 − 0.425i)14-s + (−0.885 + 0.464i)15-s + (−0.0617 + 0.189i)16-s + (−0.104 + 0.0165i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.572541 - 0.517545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.572541 - 0.517545i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.03 + 1.38i)T \) |
| 5 | \( 1 + (2.01 + 0.962i)T \) |
| good | 2 | \( 1 + (0.685 + 0.349i)T + (1.17 + 1.61i)T^{2} \) |
| 7 | \( 1 + (-1.53 - 1.53i)T + 7iT^{2} \) |
| 11 | \( 1 + (-4.90 + 1.59i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.29 - 2.54i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.429 - 0.0680i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (0.215 - 0.297i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.51 - 4.93i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.629i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (7.68 + 5.58i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.68 + 1.36i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.75 - 1.22i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (1.30 - 1.30i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.03 - 6.51i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-1.97 - 0.312i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (1.69 - 5.21i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.56 - 10.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-0.319 - 2.01i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (6.93 + 9.54i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (11.0 + 5.65i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-0.932 - 1.28i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (1.52 + 9.62i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-2.35 - 7.24i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.72 - 0.431i)T + (92.2 + 29.9i)T^{2} \) |
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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
−14.40823119662555679721638835029, −13.29080553603421616086902800272, −11.82669441075366225879737729277, −11.35544938165512993804881454021, −9.232477984224873182867906576993, −8.806297744718306929428207349362, −7.63340602089031222853346867452, −5.94977603306474409291102043670, −4.03690139876800973599972386641, −1.52137314880172834937092376119,
3.59345474141784736596862178897, 4.42753941035009730574710756613, 7.04343806738025275992851818522, 8.084187918742569129110479593008, 8.937565004226392577049454372040, 10.21046607417400607457144581581, 11.31099148033289744461080535409, 12.62479686426099848529242330272, 14.10700910503864251124246329482, 14.78884576357879577123145314265
