| L(s) = 1 | + (1.22 + 0.707i)2-s + (1.91 + 2.30i)3-s + (0.999 + 1.73i)4-s + (−1.93 − 1.11i)5-s + (0.716 + 4.18i)6-s + (6.69 + 2.05i)7-s + 2.82i·8-s + (−1.64 + 8.84i)9-s + (−1.58 − 2.73i)10-s + (−5.25 + 3.03i)11-s + (−2.07 + 5.62i)12-s + 4.30·13-s + (6.74 + 7.24i)14-s + (−1.13 − 6.61i)15-s + (−2.00 + 3.46i)16-s + (11.9 − 6.89i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.639 + 0.769i)3-s + (0.249 + 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.119 + 0.696i)6-s + (0.956 + 0.293i)7-s + 0.353i·8-s + (−0.183 + 0.983i)9-s + (−0.158 − 0.273i)10-s + (−0.477 + 0.275i)11-s + (−0.173 + 0.469i)12-s + 0.331·13-s + (0.481 + 0.517i)14-s + (−0.0755 − 0.440i)15-s + (−0.125 + 0.216i)16-s + (0.702 − 0.405i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0645 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0645 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.89160 + 1.77325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89160 + 1.77325i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (-1.91 - 2.30i)T \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| 7 | \( 1 + (-6.69 - 2.05i)T \) |
| good | 11 | \( 1 + (5.25 - 3.03i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 4.30T + 169T^{2} \) |
| 17 | \( 1 + (-11.9 + 6.89i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (5.01 - 8.68i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (12.6 + 7.27i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 50.2iT - 841T^{2} \) |
| 31 | \( 1 + (-4.34 - 7.52i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-30.8 + 53.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 5.51iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.89T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-19.8 - 11.4i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (23.1 - 13.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-75.3 + 43.4i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-2.68 + 4.64i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (59.4 + 102. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 22.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (40.7 + 70.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.9 + 86.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (76.8 + 44.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 130.T + 9.40e3T^{2} \) |
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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
−12.41184030277782668126086213460, −11.46930741074485405389769413091, −10.50481528256880270634588569174, −9.304307440379390771005450914116, −8.120719587001367765120788556730, −7.70052156060080367535412437305, −5.82658963892673535263120916554, −4.78350845298350282739529181682, −3.88717759210712431075042739367, −2.36185376033702514199823973902,
1.34120671832088161887022962513, 2.86278129823583536421133849214, 4.08517269388607931566933795572, 5.53866930910469983589642509227, 6.86835783681898884471363563485, 7.85481949504608884520113876154, 8.671891093794164173100755955899, 10.18314937818870866747489806175, 11.20041947870998277419327770504, 11.97342492866623986051292498308
