| L(s) = 1 | + (1.39 − 0.223i)2-s + (−0.755 + 0.654i)3-s + (1.89 − 0.625i)4-s + (−0.907 + 0.130i)5-s + (−0.908 + 1.08i)6-s + (−1.67 + 3.67i)7-s + (2.51 − 1.29i)8-s + (0.142 − 0.989i)9-s + (−1.23 + 0.385i)10-s + (−1.75 + 5.96i)11-s + (−1.02 + 1.71i)12-s + (0.258 − 0.117i)13-s + (−1.52 + 5.51i)14-s + (0.600 − 0.692i)15-s + (3.21 − 2.37i)16-s + (3.14 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (0.987 − 0.158i)2-s + (−0.436 + 0.378i)3-s + (0.949 − 0.312i)4-s + (−0.405 + 0.0583i)5-s + (−0.370 + 0.442i)6-s + (−0.634 + 1.39i)7-s + (0.888 − 0.458i)8-s + (0.0474 − 0.329i)9-s + (−0.391 + 0.121i)10-s + (−0.527 + 1.79i)11-s + (−0.296 + 0.495i)12-s + (0.0716 − 0.0327i)13-s + (−0.406 + 1.47i)14-s + (0.154 − 0.178i)15-s + (0.804 − 0.593i)16-s + (0.762 + 0.490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53632 + 1.17990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53632 + 1.17990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.223i)T \) |
| 3 | \( 1 + (0.755 - 0.654i)T \) |
| 23 | \( 1 + (-4.60 + 1.34i)T \) |
| good | 5 | \( 1 + (0.907 - 0.130i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (1.67 - 3.67i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.75 - 5.96i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.258 + 0.117i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.14 - 2.02i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.0815 - 0.126i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-4.40 + 6.85i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (0.844 - 0.974i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (3.53 + 0.507i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.27 - 8.85i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-4.63 + 4.01i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + (-2.73 - 1.24i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-13.0 + 5.95i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (8.58 + 7.43i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.35 - 8.01i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-8.58 + 2.52i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-2.88 + 1.85i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.819 + 1.79i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (14.2 + 2.05i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-4.87 - 5.62i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.04 + 7.28i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.23834146684853705380223704553, −10.05292113752158755720358715201, −9.653396146576619956168592489480, −8.173798651046683443377948105488, −7.08772705175870833787376841605, −6.15652729445425145283312808491, −5.28999115770168008828751281396, −4.47425907148695870945731933207, −3.24574427925411913926022095548, −2.12879494408879344495153611399,
0.876639642530133937563115007904, 3.07272077517018284065548352552, 3.76600433800400280896570355252, 5.06493360674656980132519273196, 5.94597327209768698200503501312, 6.92260086107195009623276608419, 7.55202836742450741022880905232, 8.517335533805162535949788176530, 10.14052788273665675654566460794, 10.89371286290608432767800752012
