| L(s) = 1 | + (−0.891 + 0.453i)2-s + (0.983 − 1.42i)3-s + (0.587 − 0.809i)4-s + (1.04 + 1.97i)5-s + (−0.228 + 1.71i)6-s + (0.462 − 0.462i)7-s + (−0.156 + 0.987i)8-s + (−1.06 − 2.80i)9-s + (−1.82 − 1.28i)10-s + (2.73 + 0.888i)11-s + (−0.575 − 1.63i)12-s + (1.97 − 3.86i)13-s + (−0.202 + 0.621i)14-s + (3.84 + 0.451i)15-s + (−0.309 − 0.951i)16-s + (−5.75 − 0.911i)17-s + ⋯ |
| L(s) = 1 | + (−0.630 + 0.321i)2-s + (0.567 − 0.823i)3-s + (0.293 − 0.404i)4-s + (0.467 + 0.883i)5-s + (−0.0932 + 0.700i)6-s + (0.174 − 0.174i)7-s + (−0.0553 + 0.349i)8-s + (−0.355 − 0.934i)9-s + (−0.578 − 0.406i)10-s + (0.824 + 0.267i)11-s + (−0.166 − 0.471i)12-s + (0.546 − 1.07i)13-s + (−0.0539 + 0.166i)14-s + (0.993 + 0.116i)15-s + (−0.0772 − 0.237i)16-s + (−1.39 − 0.221i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07949 - 0.0887290i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07949 - 0.0887290i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.891 - 0.453i)T \) |
| 3 | \( 1 + (-0.983 + 1.42i)T \) |
| 5 | \( 1 + (-1.04 - 1.97i)T \) |
| good | 7 | \( 1 + (-0.462 + 0.462i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.73 - 0.888i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.97 + 3.86i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (5.75 + 0.911i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-4.28 - 5.89i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.316 + 0.622i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (2.60 + 1.89i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.82 - 3.50i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.93 + 0.988i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (6.34 - 2.06i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (5.08 + 5.08i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.474 - 2.99i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (2.23 - 0.353i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.66 - 8.19i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.77 + 8.55i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.22 - 14.0i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-7.15 + 9.84i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.498 + 0.254i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-6.00 + 8.25i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.742 + 4.68i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.78 + 5.49i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-5.26 + 0.833i)T + (92.2 - 29.9i)T^{2} \) |
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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
−13.21678876168860398403964303103, −11.92060862362845050091378348934, −10.85490822525000572950599761846, −9.780535066758267711840594318982, −8.752883581796873580993637942441, −7.64980210951891826479348733876, −6.79502221983383947794434463393, −5.82677809182567896420708950392, −3.39675268561361150040740909875, −1.75720838939346191355360610197,
1.96710785256922718132988081386, 3.83031703548466779736714558086, 5.05902742651480412929057198393, 6.73461513981279225442306887817, 8.406819397621946007793338632245, 9.089490135494493837465201672126, 9.573725193808794318852609604929, 11.04222032330474261786185648316, 11.67707618938336797179694050763, 13.23143989941045319916132024586
