L(s) = 1 | + (−1.16 + 2.01i)2-s − 1.27·3-s + (−1.69 − 2.93i)4-s + (2.05 − 3.55i)5-s + (1.48 − 2.56i)6-s + (−0.208 − 2.63i)7-s + 3.23·8-s − 1.36·9-s + (4.76 + 8.25i)10-s + (−1.36 + 2.36i)11-s + (2.16 + 3.75i)12-s + (2.05 − 3.55i)13-s + (5.54 + 2.64i)14-s + (−2.61 + 4.53i)15-s + (−0.360 + 0.624i)16-s + 1.24·17-s + ⋯ |
L(s) = 1 | + (−0.820 + 1.42i)2-s − 0.737·3-s + (−0.847 − 1.46i)4-s + (0.917 − 1.58i)5-s + (0.605 − 1.04i)6-s + (−0.0789 − 0.996i)7-s + 1.14·8-s − 0.456·9-s + (1.50 + 2.60i)10-s + (−0.412 + 0.713i)11-s + (0.625 + 1.08i)12-s + (0.569 − 0.986i)13-s + (1.48 + 0.706i)14-s + (−0.676 + 1.17i)15-s + (−0.0901 + 0.156i)16-s + 0.302·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.529041 - 0.0732882i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.529041 - 0.0732882i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.208 + 2.63i)T \) |
| 19 | \( 1 + (4.34 + 0.339i)T \) |
good | 2 | \( 1 + (1.16 - 2.01i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + (-2.05 + 3.55i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.05 + 3.55i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 1.24T + 17T^{2} \) |
| 23 | \( 1 - 5.63T + 23T^{2} \) |
| 29 | \( 1 + (-0.386 + 0.669i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.21 - 2.11i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.32 + 2.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.41 - 4.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.250 + 0.434i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.00235T + 47T^{2} \) |
| 53 | \( 1 + (-0.280 - 0.485i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.61T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + (4.70 + 8.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.28 - 9.14i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + (0.675 - 1.16i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.83T + 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 + (-1.72 - 2.99i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−13.24294988989198606206992908160, −12.53843400095987329680641795842, −10.75121288514408208183277255650, −9.836199898554764562327304919136, −8.804204364972950380621275941300, −7.970444161792871124326352868422, −6.60641119191435892258125879673, −5.58213036173515321544890304138, −4.85113713332117113547854454375, −0.802541341592220561337579294404,
2.21308869103481245661817628151, 3.21255923565107538142613056924, 5.70952394450989302493954391839, 6.59288062740105341781257303069, 8.537091325016231210903393437425, 9.442588985094819343456381296663, 10.59875838895512652372023038635, 11.06882070731908241986177455782, 11.75308655083556624102569435698, 12.97103781302439113454228674620