| L(s) = 1 | + (1.74 − 1.46i)2-s + (−0.485 + 2.75i)4-s + (−1.27 − 7.22i)5-s + (13.1 − 22.6i)7-s + (12.3 + 21.3i)8-s + (−12.8 − 10.7i)10-s + (−7.39 − 12.8i)11-s + (22.9 − 8.37i)13-s + (−10.3 − 58.8i)14-s + (31.7 + 11.5i)16-s + (25.4 − 21.3i)17-s + (−18.8 − 80.6i)19-s + 20.5·20-s + (−31.6 − 11.5i)22-s + (35.7 − 202. i)23-s + ⋯ |
| L(s) = 1 | + (0.617 − 0.518i)2-s + (−0.0606 + 0.344i)4-s + (−0.113 − 0.646i)5-s + (0.707 − 1.22i)7-s + (0.544 + 0.942i)8-s + (−0.405 − 0.340i)10-s + (−0.202 − 0.350i)11-s + (0.490 − 0.178i)13-s + (−0.198 − 1.12i)14-s + (0.496 + 0.180i)16-s + (0.362 − 0.304i)17-s + (−0.227 − 0.973i)19-s + 0.229·20-s + (−0.307 − 0.111i)22-s + (0.324 − 1.83i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.86911 - 1.50644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86911 - 1.50644i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (18.8 + 80.6i)T \) |
| good | 2 | \( 1 + (-1.74 + 1.46i)T + (1.38 - 7.87i)T^{2} \) |
| 5 | \( 1 + (1.27 + 7.22i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-13.1 + 22.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (7.39 + 12.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-22.9 + 8.37i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-25.4 + 21.3i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-35.7 + 202. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-142. - 119. i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (70.1 - 121. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-3.41 - 1.24i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-90.2 - 511. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-6.20 - 5.20i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (84.3 - 478. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (283. - 237. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-53.5 + 303. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-444. - 372. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (30.3 + 172. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-936. - 341. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (616. + 224. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-138. + 239. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (731. - 266. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (949. - 796. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−12.30217936452715726113252570546, −11.02979229576594570837204404777, −10.61543273335738672809316954406, −8.791883020166161735496670761502, −8.085299957306324919520651473942, −6.88210788351583002123729865469, −4.98322074608327146071436389972, −4.36454776008643194895965077973, −2.96971005842290651594297963268, −1.02423443874333882585128087761,
1.82044087689850393958038139247, 3.66723451845734580602281099593, 5.17932272058319521237473435734, 5.89655813125327378809834879221, 7.08617119625021091898523851378, 8.259262531968492572622514511855, 9.520766388743377265150887864871, 10.57895216186872286384349394496, 11.60893208860626142123961748823, 12.58213380033808421420751792278
