L(s) = 1 | + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (2.03 + 0.917i)5-s + (0.309 + 0.951i)6-s − 4.80i·7-s + (0.951 − 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.456 − 2.18i)10-s + (0.714 − 0.518i)11-s + (0.587 − 0.809i)12-s + (1.66 − 2.28i)13-s + (−3.88 + 2.82i)14-s + (−1.65 − 1.50i)15-s + (−0.809 − 0.587i)16-s + (−1.57 + 0.512i)17-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.572i)2-s + (−0.549 − 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.912 + 0.410i)5-s + (0.126 + 0.388i)6-s − 1.81i·7-s + (0.336 − 0.109i)8-s + (0.269 + 0.195i)9-s + (−0.144 − 0.692i)10-s + (0.215 − 0.156i)11-s + (0.169 − 0.233i)12-s + (0.460 − 0.633i)13-s + (−1.03 + 0.755i)14-s + (−0.427 − 0.387i)15-s + (−0.202 − 0.146i)16-s + (−0.382 + 0.124i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.684296 - 0.546488i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.684296 - 0.546488i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.587 + 0.809i)T \) |
| 3 | \( 1 + (0.951 + 0.309i)T \) |
| 5 | \( 1 + (-2.03 - 0.917i)T \) |
good | 7 | \( 1 + 4.80iT - 7T^{2} \) |
| 11 | \( 1 + (-0.714 + 0.518i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.66 + 2.28i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.57 - 0.512i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.66 + 5.11i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.44 - 4.73i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.10 - 3.40i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.22 - 9.93i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (1.02 - 1.41i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.40 - 1.01i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.27iT - 43T^{2} \) |
| 47 | \( 1 + (8.29 + 2.69i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.37 + 1.09i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.37 - 6.08i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.0697 + 0.0506i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (11.3 - 3.69i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.08 - 3.33i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.24 - 5.84i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.88 + 11.9i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (12.6 - 4.10i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-15.1 + 10.9i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-17.3 - 5.64i)T + (78.4 + 57.0i)T^{2} \) |
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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
−13.12269424889476471201229767950, −11.42815315736287151260344632351, −10.66405445554444785840531750309, −10.16621043982084590843709450452, −8.862698797747793053735989363724, −7.31993191135650874057816157105, −6.58296775625480162173867053109, −4.91337899669824332755938034696, −3.33495566488987431538765481070, −1.23334448227241173793524948406,
2.08030982605615545982831490121, 4.71006397044532192505715661509, 5.90441883007124945679377650570, 6.35746047367527950596297032892, 8.282794610117585009581090768757, 9.145024981551667692816311991064, 9.845145002999002364443474160857, 11.22315859791817778019525790502, 12.23832312293567736505971817238, 13.14182453488250183004207302168