L(s) = 1 | + (0.585 − 0.156i)2-s + (4.00 + 1.07i)3-s + (−3.14 + 1.81i)4-s + (−4.51 − 2.15i)5-s + 2.51·6-s + (3.65 − 5.97i)7-s + (−3.27 + 3.27i)8-s + (7.05 + 4.07i)9-s + (−2.98 − 0.552i)10-s + (−2.32 − 4.01i)11-s + (−14.5 + 3.89i)12-s + (−6.67 + 6.67i)13-s + (1.20 − 4.07i)14-s + (−15.7 − 13.4i)15-s + (5.86 − 10.1i)16-s + (−4.93 + 18.4i)17-s + ⋯ |
L(s) = 1 | + (0.292 − 0.0784i)2-s + (1.33 + 0.357i)3-s + (−0.786 + 0.454i)4-s + (−0.902 − 0.430i)5-s + 0.418·6-s + (0.521 − 0.853i)7-s + (−0.408 + 0.408i)8-s + (0.784 + 0.452i)9-s + (−0.298 − 0.0552i)10-s + (−0.210 − 0.365i)11-s + (−1.21 + 0.324i)12-s + (−0.513 + 0.513i)13-s + (0.0857 − 0.290i)14-s + (−1.04 − 0.896i)15-s + (0.366 − 0.634i)16-s + (−0.290 + 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.29473 + 0.0952701i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29473 + 0.0952701i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (4.51 + 2.15i)T \) |
| 7 | \( 1 + (-3.65 + 5.97i)T \) |
good | 2 | \( 1 + (-0.585 + 0.156i)T + (3.46 - 2i)T^{2} \) |
| 3 | \( 1 + (-4.00 - 1.07i)T + (7.79 + 4.5i)T^{2} \) |
| 11 | \( 1 + (2.32 + 4.01i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (6.67 - 6.67i)T - 169iT^{2} \) |
| 17 | \( 1 + (4.93 - 18.4i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-22.5 - 13.0i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (2.06 + 7.71i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 39.4iT - 841T^{2} \) |
| 31 | \( 1 + (-14.1 - 24.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-33.9 + 9.10i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 20.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (14.8 - 14.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (57.3 - 15.3i)T + (1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (31.9 + 8.55i)T + (2.43e3 + 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-36.4 + 21.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-19.6 + 33.9i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (9.45 - 35.2i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 41.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (6.72 + 1.80i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (119. + 68.8i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (37.7 - 37.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-106. - 61.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (4.90 + 4.90i)T + 9.40e3iT^{2} \) |
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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
−16.22904982085593712384455487927, −14.82880431769507557900409568082, −14.07056794213760300324875265267, −13.05745602064480985529531257398, −11.64996728762659879520688229247, −9.780302074898498937465228306804, −8.416446194419154850474205303365, −7.82041467013215021501110357924, −4.54715564077484876907490993728, −3.53102191252042823505587009593,
3.00130932937809305390410351220, 4.96995080343814645812913404600, 7.38475076429070677827271183695, 8.532341658092925093621316331822, 9.645958598264613854118195010365, 11.61831002023069832398638134997, 12.99946321674829362973145280426, 14.14144542609584342269077251348, 14.92147886264218814325270058738, 15.63856371241578516398605130160