| L(s) = 1 | + (−1.03 − 0.0909i)2-s + (−0.896 − 0.158i)4-s + (−2.23 + 0.102i)5-s + (−0.567 − 0.397i)7-s + (2.93 + 0.786i)8-s + (2.33 + 0.0969i)10-s + (−0.597 + 1.64i)11-s + (−0.0630 − 0.720i)13-s + (0.554 + 0.464i)14-s + (−1.26 − 0.461i)16-s + (2.95 − 0.792i)17-s + (3.38 − 1.95i)19-s + (2.01 + 0.261i)20-s + (0.770 − 1.65i)22-s + (4.63 + 6.61i)23-s + ⋯ |
| L(s) = 1 | + (−0.735 − 0.0643i)2-s + (−0.448 − 0.0790i)4-s + (−0.998 + 0.0456i)5-s + (−0.214 − 0.150i)7-s + (1.03 + 0.277i)8-s + (0.737 + 0.0306i)10-s + (−0.180 + 0.495i)11-s + (−0.0174 − 0.199i)13-s + (0.148 + 0.124i)14-s + (−0.317 − 0.115i)16-s + (0.716 − 0.192i)17-s + (0.777 − 0.448i)19-s + (0.451 + 0.0584i)20-s + (0.164 − 0.352i)22-s + (0.965 + 1.37i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.649808 + 0.0277868i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.649808 + 0.0277868i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.23 - 0.102i)T \) |
| good | 2 | \( 1 + (1.03 + 0.0909i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (0.567 + 0.397i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.597 - 1.64i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0630 + 0.720i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.95 + 0.792i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.38 + 1.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.63 - 6.61i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.68 + 4.77i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.764 + 4.33i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.86 - 10.6i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.71 - 3.23i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.74 - 5.87i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.32 + 3.32i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-5.90 + 5.90i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.63 - 0.959i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 6.44i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 0.978i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (5.58 + 3.22i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.16 - 8.08i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.04 + 1.24i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.156 - 1.78i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (2.78 + 4.81i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.20 - 1.95i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.27586023151147915813994150605, −10.06448248045731005976874547846, −9.609124563283860031144570867925, −8.425728737698072769141633971088, −7.75546726614370325621765467915, −6.96153499720665691228133280717, −5.29507834600166099606650336904, −4.36827080755192148392650308165, −3.10280211983640254994549873795, −0.948212936138122876615096375239,
0.836648713086775467802123676355, 3.14296163813591514569906186057, 4.26162060314389552980248565706, 5.35555363998229496863655362917, 6.88648676425763859345633962354, 7.73149999826207411182019595780, 8.567370037410931103855373040793, 9.172290303526708561326681896931, 10.38603103245768137769846018649, 10.95299382535760314015916359598
