L(s) = 1 | + (−1.34 + 0.425i)2-s + (1.03 − 1.38i)3-s + (1.63 − 1.14i)4-s + (3.18 + 0.952i)5-s + (−0.807 + 2.31i)6-s + (−0.256 + 0.594i)7-s + (−1.72 + 2.24i)8-s + (−0.852 − 2.87i)9-s + (−4.69 + 0.0682i)10-s + (0.966 − 4.07i)11-s + (0.105 − 3.46i)12-s + (3.56 + 5.42i)13-s + (0.0930 − 0.910i)14-s + (4.61 − 3.42i)15-s + (1.36 − 3.75i)16-s + (1.51 + 0.552i)17-s + ⋯ |
L(s) = 1 | + (−0.953 + 0.300i)2-s + (0.598 − 0.801i)3-s + (0.819 − 0.573i)4-s + (1.42 + 0.426i)5-s + (−0.329 + 0.944i)6-s + (−0.0969 + 0.224i)7-s + (−0.608 + 0.793i)8-s + (−0.284 − 0.958i)9-s + (−1.48 + 0.0215i)10-s + (0.291 − 1.22i)11-s + (0.0303 − 0.999i)12-s + (0.989 + 1.50i)13-s + (0.0248 − 0.243i)14-s + (1.19 − 0.885i)15-s + (0.342 − 0.939i)16-s + (0.368 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55267 - 0.258555i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55267 - 0.258555i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 - 0.425i)T \) |
| 3 | \( 1 + (-1.03 + 1.38i)T \) |
good | 5 | \( 1 + (-3.18 - 0.952i)T + (4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.256 - 0.594i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-0.966 + 4.07i)T + (-9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-3.56 - 5.42i)T + (-5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-1.51 - 0.552i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.333 - 0.914i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (0.276 + 0.641i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (8.74 + 0.509i)T + (28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (-5.76 - 0.673i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (0.420 + 0.501i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (1.61 + 0.813i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (2.18 + 2.06i)T + (2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (5.52 - 0.645i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (-6.83 + 3.94i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.723 + 3.05i)T + (-52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (0.304 - 0.226i)T + (17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (11.6 - 0.680i)T + (66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (1.09 + 6.21i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (0.944 - 5.35i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-14.2 + 7.14i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (-4.78 - 9.53i)T + (-49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (-3.19 + 18.0i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (0.714 + 2.38i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−10.28942057451296852014185947411, −9.267608639839415686314533430705, −8.964866276758962098132370187636, −8.039046454053894614148236293730, −6.86090694199842132575341488026, −6.24204865145540294739652956179, −5.73393364916229348426038826604, −3.43479248562852880900897924659, −2.21017364378575738691943265009, −1.36080067555393346064447408582,
1.47313326516766178978819395785, 2.59933586766218508084425804699, 3.74385454377749281435845775311, 5.16779870227306954003888903527, 6.09455540253531791053228842876, 7.37702485859385594408918506798, 8.282340882099689151681454844897, 9.125760770836706893187164357871, 9.781893512614389456638616286965, 10.22023756070826373653096684686