L(s) = 1 | + (−0.190 + 0.330i)2-s + (−1.11 − 1.93i)3-s + (0.927 + 1.60i)4-s + (−1.11 + 1.93i)5-s + 0.854·6-s + (2 + 1.73i)7-s − 1.47·8-s + (−1 + 1.73i)9-s + (−0.427 − 0.739i)10-s + (−1.5 − 2.59i)11-s + (2.07 − 3.59i)12-s + (−0.954 + 0.330i)14-s + 5.00·15-s + (−1.57 + 2.72i)16-s + (3.73 + 6.47i)17-s + (−0.381 − 0.661i)18-s + ⋯ |
L(s) = 1 | + (−0.135 + 0.233i)2-s + (−0.645 − 1.11i)3-s + (0.463 + 0.802i)4-s + (−0.499 + 0.866i)5-s + 0.348·6-s + (0.755 + 0.654i)7-s − 0.520·8-s + (−0.333 + 0.577i)9-s + (−0.135 − 0.233i)10-s + (−0.452 − 0.783i)11-s + (0.598 − 1.03i)12-s + (−0.255 + 0.0884i)14-s + 1.29·15-s + (−0.393 + 0.681i)16-s + (0.906 + 1.56i)17-s + (−0.0900 − 0.155i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9475417208\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9475417208\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2 - 1.73i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.190 - 0.330i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.11 + 1.93i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.11 - 1.93i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.73 - 6.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 2.59i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.88 + 3.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.35 - 7.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-0.736 + 1.27i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.736 + 1.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.73 - 6.47i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + (-5.35 - 9.27i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.35 - 9.27i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.11 - 1.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−10.33336830098624025099571754029, −8.677064286733530576401204518787, −8.216190210360653738104460037450, −7.47930575835672689317093316523, −6.79435386943882934250955087155, −6.12357493532839406634008462336, −5.22700521423171777629234952259, −3.61390989824593194726994036616, −2.78711187513178353260298122715, −1.52803282246173312726481778507,
0.46928506186351079857578087747, 1.78174561676838301276531874606, 3.46850859200000033925920931681, 4.65280073881114745836965557753, 5.03946911914954901355657644264, 5.71255287163473565103921145306, 7.20671776803264796450101768412, 7.74901993354293590247650011547, 9.023238940327093346007563680649, 9.834802311199408975139238633519