L(s) = 1 | + (−0.5 + 1.53i)2-s + (0.809 − 0.587i)3-s + (−0.5 − 0.363i)4-s + (0.190 + 0.587i)5-s + (0.5 + 1.53i)6-s + (0.809 + 0.587i)7-s + (−1.80 + 1.31i)8-s + (0.309 − 0.951i)9-s − 10-s + (2.80 + 1.76i)11-s − 0.618·12-s + (−0.0729 + 0.224i)13-s + (−1.30 + 0.951i)14-s + (0.5 + 0.363i)15-s + (−1.50 − 4.61i)16-s + (1.69 + 5.20i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 1.08i)2-s + (0.467 − 0.339i)3-s + (−0.250 − 0.181i)4-s + (0.0854 + 0.262i)5-s + (0.204 + 0.628i)6-s + (0.305 + 0.222i)7-s + (−0.639 + 0.464i)8-s + (0.103 − 0.317i)9-s − 0.316·10-s + (0.846 + 0.531i)11-s − 0.178·12-s + (−0.0202 + 0.0622i)13-s + (−0.349 + 0.254i)14-s + (0.129 + 0.0937i)15-s + (−0.375 − 1.15i)16-s + (0.410 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.854390 + 0.957320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.854390 + 0.957320i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-2.80 - 1.76i)T \) |
good | 2 | \( 1 + (0.5 - 1.53i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.190 - 0.587i)T + (-4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (0.0729 - 0.224i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.69 - 5.20i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2 + 1.45i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 + (3.42 + 2.48i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.927 + 2.85i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.85 + 3.52i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.54 + 1.84i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 + (-9.59 + 6.96i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.33 + 7.19i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.92 - 1.40i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.07 - 9.45i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 + (4.16 + 12.8i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (7.78 + 5.65i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.57 + 4.84i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (0.145 + 0.449i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + (5.78 - 17.7i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.34863743318889450248517413838, −11.73337021615127725125844523383, −10.28691015887542513831974395420, −9.167679652138246790770490067555, −8.333594927742506886912413292225, −7.48651650974909689029648566979, −6.57425292831900081127331056514, −5.65999297412013989086556398345, −3.90439555524014659031966272707, −2.16036572573081923361759851181,
1.34198471512577951866034080089, 2.92988363439295471638258207660, 4.02996426462646267312076079424, 5.56624524916865205402574191144, 7.03448817445642150582966527089, 8.383139908639715914282471479415, 9.311535848546917941859511877668, 9.967485312777577342558941487169, 10.99948900618568227910255342625, 11.76395015109755032290323997108