L(s) = 1 | + (0.5 − 0.866i)3-s + (−2.12 + 1.22i)5-s + (2.63 + 0.272i)7-s + (−0.499 − 0.866i)9-s + (1.09 + 0.632i)11-s − 2.99i·13-s + 2.45i·15-s + (1.58 + 0.916i)17-s + (2.07 + 3.60i)19-s + (1.55 − 2.14i)21-s + (−5.83 + 3.36i)23-s + (0.507 − 0.879i)25-s − 0.999·27-s + 9.42·29-s + (4.71 − 8.17i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.949 + 0.548i)5-s + (0.994 + 0.102i)7-s + (−0.166 − 0.288i)9-s + (0.330 + 0.190i)11-s − 0.831i·13-s + 0.633i·15-s + (0.385 + 0.222i)17-s + (0.477 + 0.826i)19-s + (0.338 − 0.467i)21-s + (−1.21 + 0.702i)23-s + (0.101 − 0.175i)25-s − 0.192·27-s + 1.74·29-s + (0.847 − 1.46i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.773257401\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.773257401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.63 - 0.272i)T \) |
good | 5 | \( 1 + (2.12 - 1.22i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.09 - 0.632i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.99iT - 13T^{2} \) |
| 17 | \( 1 + (-1.58 - 0.916i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.07 - 3.60i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.83 - 3.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + (-4.71 + 8.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.75 - 6.50i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.08iT - 41T^{2} \) |
| 43 | \( 1 + 6.27iT - 43T^{2} \) |
| 47 | \( 1 + (-3.67 - 6.36i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0358 - 0.0620i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.68 - 2.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.61 + 5.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.43 - 1.40i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.92iT - 71T^{2} \) |
| 73 | \( 1 + (-7.01 - 4.05i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.54 - 0.891i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.33T + 83T^{2} \) |
| 89 | \( 1 + (-7.42 + 4.28i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.10iT - 97T^{2} \) |
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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
−9.708516430749395766553290307785, −8.379081436389082214042813122111, −7.957850468935611560790175288097, −7.49655982020276269235224724424, −6.39209352450478712351898729651, −5.51934848959339147873484754015, −4.32511868437207826049462075861, −3.51314589901773771491017317930, −2.41370248891671045366736384252, −1.08268385656512724001967166905,
0.942995223952753354434736400514, 2.45188172460304981346618351934, 3.75244343274722570139304533686, 4.52366892486309111196842902895, 5.00901916636968277151446818179, 6.36652814532607922511682357412, 7.35456592872925510104374079881, 8.272318608624875762513870719149, 8.561361316579968585376805612238, 9.514066433857221133931127714904