L(s) = 1 | + (1.90 + 0.618i)2-s + (3.23 + 2.35i)4-s + 5.25·7-s + (4.70 + 6.47i)8-s + 19.9i·11-s − 8.47i·13-s + (9.99 + 3.24i)14-s + (4.94 + 15.2i)16-s + 11.8i·17-s − 15.2i·19-s + (−12.3 + 37.8i)22-s + 0.555·23-s + (5.23 − 16.1i)26-s + (17.0 + 12.3i)28-s − 10.9·29-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)2-s + (0.809 + 0.587i)4-s + 0.751·7-s + (0.587 + 0.809i)8-s + 1.81i·11-s − 0.651i·13-s + (0.714 + 0.232i)14-s + (0.309 + 0.951i)16-s + 0.699i·17-s − 0.800i·19-s + (−0.559 + 1.72i)22-s + 0.0241·23-s + (0.201 − 0.619i)26-s + (0.607 + 0.441i)28-s − 0.377·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.779855940\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.779855940\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.90 - 0.618i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5.25T + 49T^{2} \) |
| 11 | \( 1 - 19.9iT - 121T^{2} \) |
| 13 | \( 1 + 8.47iT - 169T^{2} \) |
| 17 | \( 1 - 11.8iT - 289T^{2} \) |
| 19 | \( 1 + 15.2iT - 361T^{2} \) |
| 23 | \( 1 - 0.555T + 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 - 18.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 53.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 66.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 50.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.55iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 92.8iT - 9.40e3T^{2} \) |
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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
−10.33909230429730573478256448048, −9.213844501398420120631394238282, −8.088840597145716300610625249446, −7.45399530164737139664782558447, −6.68160140080564829861013407368, −5.56294086276630935004716217951, −4.76090063325721043735612083329, −4.07126010620538566974582117213, −2.68885360663636683419919397043, −1.66945626997710078208796103136,
0.925523787964879245609929429359, 2.23409898944043421477932975678, 3.40031763408424442854448448245, 4.25546868738986505374531154502, 5.37517300367809494157630871347, 5.95509516065862680020859155861, 7.00431669259741299257078399355, 7.975621320326049955620661430256, 8.878465570317507738376793037620, 9.914152352718175529950126013426