| L(s) = 1 | + (0.0137 − 0.263i)2-s + (1.98 − 1.61i)3-s + (1.92 + 0.201i)4-s + (−2.07 − 0.827i)5-s + (−0.396 − 0.545i)6-s + (0.819 + 2.51i)7-s + (0.161 − 1.02i)8-s + (0.738 − 3.47i)9-s + (−0.246 + 0.535i)10-s + (−3.43 + 0.730i)11-s + (4.14 − 2.69i)12-s + (−1.66 + 3.27i)13-s + (0.673 − 0.180i)14-s + (−5.46 + 1.69i)15-s + (3.50 + 0.746i)16-s + (−5.50 + 2.11i)17-s + ⋯ |
| L(s) = 1 | + (0.00974 − 0.186i)2-s + (1.14 − 0.929i)3-s + (0.960 + 0.100i)4-s + (−0.928 − 0.370i)5-s + (−0.161 − 0.222i)6-s + (0.309 + 0.950i)7-s + (0.0572 − 0.361i)8-s + (0.246 − 1.15i)9-s + (−0.0779 + 0.169i)10-s + (−1.03 + 0.220i)11-s + (1.19 − 0.776i)12-s + (−0.462 + 0.907i)13-s + (0.179 − 0.0483i)14-s + (−1.41 + 0.438i)15-s + (0.877 + 0.186i)16-s + (−1.33 + 0.512i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50236 - 0.673394i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50236 - 0.673394i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.07 + 0.827i)T \) |
| 7 | \( 1 + (-0.819 - 2.51i)T \) |
| good | 2 | \( 1 + (-0.0137 + 0.263i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.98 + 1.61i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (3.43 - 0.730i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.66 - 3.27i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (5.50 - 2.11i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (0.817 + 7.77i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-6.38 - 0.334i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-0.147 + 0.202i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.511 + 1.14i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.68 + 4.12i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (-0.567 + 0.184i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.65 - 5.65i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.376 - 0.980i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (-1.86 - 2.30i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-3.73 + 4.15i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-2.91 + 2.62i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (0.424 + 1.10i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (-7.13 - 5.18i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (6.72 + 4.36i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (4.00 + 8.99i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (6.00 + 0.951i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (2.57 + 2.85i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 1.59i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.78986755875385314708824795007, −11.64596831773383210183193116065, −11.00693754893750349497835385642, −9.111300969033487186303520164629, −8.452654294262046481744445472783, −7.41980736128122310147331699113, −6.75370365906056691324993259371, −4.78743222110548735906069863889, −2.91477126902626649558349371697, −2.07666752087331681274864109701,
2.69885460069531849061566509644, 3.67699850246493008522013770030, 5.01945333497379825398003828415, 6.93180104340063250856995395947, 7.82281515624014133728837179632, 8.479760367540502544200727618323, 10.26240709663938936798230930449, 10.52860395967876220906314304105, 11.59885948966127171743885736940, 12.98803540093078485754366507962
