| L(s) = 1 | + (−0.988 − 0.149i)2-s + (−0.826 + 0.563i)3-s + (0.955 + 0.294i)4-s + (0.0491 + 0.656i)5-s + (0.900 − 0.433i)6-s + (2.17 + 1.50i)7-s + (−0.900 − 0.433i)8-s + (0.365 − 0.930i)9-s + (0.0491 − 0.656i)10-s + (−1.06 − 2.72i)11-s + (−0.955 + 0.294i)12-s + (−3.41 + 4.28i)13-s + (−1.93 − 1.80i)14-s + (−0.410 − 0.514i)15-s + (0.826 + 0.563i)16-s + (2.53 + 2.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.105i)2-s + (−0.477 + 0.325i)3-s + (0.477 + 0.147i)4-s + (0.0219 + 0.293i)5-s + (0.367 − 0.177i)6-s + (0.823 + 0.567i)7-s + (−0.318 − 0.153i)8-s + (0.121 − 0.310i)9-s + (0.0155 − 0.207i)10-s + (−0.322 − 0.820i)11-s + (−0.275 + 0.0850i)12-s + (−0.947 + 1.18i)13-s + (−0.516 − 0.483i)14-s + (−0.105 − 0.132i)15-s + (0.206 + 0.140i)16-s + (0.614 + 0.570i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.608596 + 0.491497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.608596 + 0.491497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.988 + 0.149i)T \) |
| 3 | \( 1 + (0.826 - 0.563i)T \) |
| 7 | \( 1 + (-2.17 - 1.50i)T \) |
| good | 5 | \( 1 + (-0.0491 - 0.656i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.06 + 2.72i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (3.41 - 4.28i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-2.53 - 2.35i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.54 - 6.14i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.94 - 2.73i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (1.81 - 7.94i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.30 + 3.99i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.33 - 2.26i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (3.41 + 1.64i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.23 + 1.55i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-5.41 - 0.815i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-0.544 - 0.167i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.888 + 11.8i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-2.08 + 0.643i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.93 + 6.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.60 + 11.4i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.77 - 0.719i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (7.88 + 13.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.90 - 12.4i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (1.69 - 4.32i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 9.63T + 97T^{2} \) |
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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
−11.87785314044551058326382446017, −10.95714766350599211566127161979, −10.17793596451964081948016020386, −9.211333162086691440006102627718, −8.238964487991569147036392884327, −7.28239866513039242304723005137, −6.01210299239566077002104379041, −5.06546224952353931902333156186, −3.45927987123889447369541164925, −1.76755933375412885809420343884,
0.808108047294679830369884801569, 2.55061829891538161842048308863, 4.70913137681304375204425946459, 5.47073552154705684322068768587, 7.13863403796297083820624727720, 7.51927982109629653062818296780, 8.599707091353148848578535091571, 9.906001235606887703484415614628, 10.46146607972460645526299628500, 11.56258379858707967525952364447
