L(s) = 1 | + (−0.951 − 0.309i)2-s + (0.809 + 0.587i)4-s + (−1.95 + 1.07i)5-s − 3.03i·7-s + (−0.587 − 0.809i)8-s + (2.19 − 0.420i)10-s + (−0.791 + 2.43i)11-s + (−0.945 + 0.307i)13-s + (−0.938 + 2.88i)14-s + (0.309 + 0.951i)16-s + (0.558 + 0.769i)17-s + (−5.39 + 3.91i)19-s + (−2.21 − 0.279i)20-s + (1.50 − 2.07i)22-s + (−5.81 − 1.89i)23-s + ⋯ |
L(s) = 1 | + (−0.672 − 0.218i)2-s + (0.404 + 0.293i)4-s + (−0.876 + 0.482i)5-s − 1.14i·7-s + (−0.207 − 0.286i)8-s + (0.694 − 0.132i)10-s + (−0.238 + 0.734i)11-s + (−0.262 + 0.0852i)13-s + (−0.250 + 0.771i)14-s + (0.0772 + 0.237i)16-s + (0.135 + 0.186i)17-s + (−1.23 + 0.898i)19-s + (−0.496 − 0.0624i)20-s + (0.320 − 0.441i)22-s + (−1.21 − 0.394i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00833580 + 0.0528015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00833580 + 0.0528015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 + 0.309i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.95 - 1.07i)T \) |
good | 7 | \( 1 + 3.03iT - 7T^{2} \) |
| 11 | \( 1 + (0.791 - 2.43i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.945 - 0.307i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.558 - 0.769i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.39 - 3.91i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (5.81 + 1.89i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (7.82 + 5.68i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (5.65 - 4.10i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.94 - 0.955i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.167 + 0.516i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 3.08iT - 43T^{2} \) |
| 47 | \( 1 + (-1.62 + 2.23i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (1.08 - 1.49i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.998 - 3.07i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.68 - 11.3i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (7.32 + 10.0i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-8.64 - 6.28i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-13.7 - 4.47i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (12.3 + 8.95i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (5.22 + 7.19i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.76 + 8.50i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.835 - 1.15i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−11.29470868096408764033681750040, −10.42168474739336199560079293928, −10.05659997600013288109613653346, −8.694271334436600171350236854585, −7.68852967095948176613238897249, −7.29392620398948904884945075978, −6.16549077777200532187305455901, −4.36084802798890461206411636151, −3.65234721238212747757576351777, −2.00839954415751535522059873848,
0.03883469792385561182850671446, 2.14857547585721113246221272125, 3.62507220178608331125485580160, 5.10423784857350423108729974941, 5.95244630620577186918359140573, 7.21195411262647177940983566292, 8.128417120597269565365044105780, 8.819094634916512976430613231540, 9.463974432731671624200373139183, 10.84967674803792076094174275191