L(s) = 1 | + (−1.22 − 1.22i)3-s + (4.67 + 1.77i)5-s + (0.550 − 0.550i)7-s + 2.99i·9-s + 1.55·11-s + (9.55 + 9.55i)13-s + (−3.55 − 7.89i)15-s + (11.1 − 11.1i)17-s − 12.6i·19-s − 1.34·21-s + (21.3 + 21.3i)23-s + (18.6 + 16.5i)25-s + (3.67 − 3.67i)27-s − 44.0i·29-s + 44.4·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.934 + 0.355i)5-s + (0.0786 − 0.0786i)7-s + 0.333i·9-s + 0.140·11-s + (0.734 + 0.734i)13-s + (−0.236 − 0.526i)15-s + (0.655 − 0.655i)17-s − 0.668i·19-s − 0.0642·21-s + (0.928 + 0.928i)23-s + (0.747 + 0.663i)25-s + (0.136 − 0.136i)27-s − 1.51i·29-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.69575 - 0.111354i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69575 - 0.111354i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 + (-4.67 - 1.77i)T \) |
good | 7 | \( 1 + (-0.550 + 0.550i)T - 49iT^{2} \) |
| 11 | \( 1 - 1.55T + 121T^{2} \) |
| 13 | \( 1 + (-9.55 - 9.55i)T + 169iT^{2} \) |
| 17 | \( 1 + (-11.1 + 11.1i)T - 289iT^{2} \) |
| 19 | \( 1 + 12.6iT - 361T^{2} \) |
| 23 | \( 1 + (-21.3 - 21.3i)T + 529iT^{2} \) |
| 29 | \( 1 + 44.0iT - 841T^{2} \) |
| 31 | \( 1 - 44.4T + 961T^{2} \) |
| 37 | \( 1 + (-20.6 + 20.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 48.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.2 - 36.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (42.5 - 42.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (54.4 + 54.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 47.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 59.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (81.2 - 81.2i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 87.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-75.9 - 75.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 97.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (41.0 + 41.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 52.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (37 - 37i)T - 9.40e3iT^{2} \) |
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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
−11.67553634608602922012666876827, −11.12451470141217171000237289492, −9.909634329174639754858461434012, −9.158717830000846545173896628751, −7.77488917199181393218541443340, −6.67769851114011672269424652429, −5.91856876326702440171120733018, −4.68233255335959513206543665732, −2.88075086577379548882858167838, −1.32844154521462382965347329959,
1.29105787961853537195263083430, 3.20072044383043660764411605927, 4.75821520727822079644926806783, 5.72777998598385733828651653845, 6.58391123664760406837101209374, 8.215755654473045014079620963926, 9.043402422462823878162390797654, 10.24414647075233640779668857949, 10.64052757539304225904853281879, 12.02678972256527805745553936932