| L(s) = 1 | + (1.73 − 0.228i)2-s + (0.759 − 1.55i)3-s + (1.02 − 0.274i)4-s + (1.19 + 2.42i)5-s + (0.962 − 2.87i)6-s + (−1.42 − 0.701i)7-s + (−1.51 + 0.627i)8-s + (−1.84 − 2.36i)9-s + (2.63 + 3.94i)10-s + (−2.75 + 0.936i)11-s + (0.351 − 1.80i)12-s + (−0.0154 − 0.0576i)13-s + (−2.62 − 0.891i)14-s + (4.69 − 0.0194i)15-s + (−4.32 + 2.49i)16-s + (3.87 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (1.22 − 0.161i)2-s + (0.438 − 0.898i)3-s + (0.513 − 0.137i)4-s + (0.535 + 1.08i)5-s + (0.392 − 1.17i)6-s + (−0.537 − 0.265i)7-s + (−0.536 + 0.222i)8-s + (−0.615 − 0.788i)9-s + (0.832 + 1.24i)10-s + (−0.832 + 0.282i)11-s + (0.101 − 0.521i)12-s + (−0.00428 − 0.0159i)13-s + (−0.702 − 0.238i)14-s + (1.21 − 0.00503i)15-s + (−1.08 + 0.624i)16-s + (0.940 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98508 - 0.515555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98508 - 0.515555i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.759 + 1.55i)T \) |
| 17 | \( 1 + (-3.87 + 1.40i)T \) |
| good | 2 | \( 1 + (-1.73 + 0.228i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (-1.19 - 2.42i)T + (-3.04 + 3.96i)T^{2} \) |
| 7 | \( 1 + (1.42 + 0.701i)T + (4.26 + 5.55i)T^{2} \) |
| 11 | \( 1 + (2.75 - 0.936i)T + (8.72 - 6.69i)T^{2} \) |
| 13 | \( 1 + (0.0154 + 0.0576i)T + (-11.2 + 6.5i)T^{2} \) |
| 19 | \( 1 + (1.06 + 0.442i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.90 + 5.58i)T + (-3.00 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.539 - 8.23i)T + (-28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (0.835 - 2.46i)T + (-24.5 - 18.8i)T^{2} \) |
| 37 | \( 1 + (-0.970 - 4.87i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-10.7 + 0.707i)T + (40.6 - 5.35i)T^{2} \) |
| 43 | \( 1 + (7.37 + 5.65i)T + (11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (0.0354 - 0.132i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-3.58 + 8.65i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-0.271 + 2.06i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (3.05 - 6.18i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (-4.28 - 2.47i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.84 + 1.36i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (2.68 + 1.79i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.385 + 1.13i)T + (-62.6 + 48.0i)T^{2} \) |
| 83 | \( 1 + (0.972 + 7.38i)T + (-80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (10.1 - 10.1i)T - 89iT^{2} \) |
| 97 | \( 1 + (12.5 + 0.820i)T + (96.1 + 12.6i)T^{2} \) |
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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
−12.94924136388338199177664256098, −12.43639478553323237188281705669, −11.12619246861394545917304657884, −10.04609394025116658017026094352, −8.663481580045822763374517879668, −7.16024355808027284775464360465, −6.45256239405392576750709122602, −5.22844604743396342091036777344, −3.33847967795876715585010770314, −2.57247931055330730496443808686,
2.88718227621192413081452691379, 4.16653415977342744366127070745, 5.31030561937156228947620652540, 5.86270523247757424725467863162, 7.936927636439665485994627841322, 9.215538341040169716428701044377, 9.733733147613581671204279888503, 11.20573045905175916062706983811, 12.55582480697074876059645995565, 13.17212184839750329960652173931
