L(s) = 1 | + (0.587 + 0.809i)2-s + (0.951 + 0.309i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s − 4.47i·7-s + (2.85 − 0.927i)8-s + (0.809 + 0.587i)9-s + (−2.61 + 1.90i)11-s + (0.587 − 0.809i)12-s + (1.98 − 2.73i)13-s + (3.61 − 2.62i)14-s + (0.809 + 0.587i)16-s + (−2.71 + 0.881i)17-s + 0.999i·18-s + (1 + 3.07i)19-s + ⋯ |
L(s) = 1 | + (0.415 + 0.572i)2-s + (0.549 + 0.178i)3-s + (0.154 − 0.475i)4-s + (0.126 + 0.388i)6-s − 1.69i·7-s + (1.00 − 0.327i)8-s + (0.269 + 0.195i)9-s + (−0.789 + 0.573i)11-s + (0.169 − 0.233i)12-s + (0.551 − 0.758i)13-s + (0.966 − 0.702i)14-s + (0.202 + 0.146i)16-s + (−0.658 + 0.213i)17-s + 0.235i·18-s + (0.229 + 0.706i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.08835 - 0.118904i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.08835 - 0.118904i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.587 - 0.809i)T + (-0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + 4.47iT - 7T^{2} \) |
| 11 | \( 1 + (2.61 - 1.90i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.98 + 2.73i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.71 - 0.881i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1 - 3.07i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 3.61i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.35 - 4.16i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.23 - 6.88i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (4.75 - 6.54i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.11 - 0.812i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 5.70iT - 43T^{2} \) |
| 47 | \( 1 + (-4.97 - 1.61i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.31 - 0.427i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.23 + 2.35i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.363i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.97 + 1.61i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.236 - 0.726i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.81 + 2.5i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.35 - 1.09i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.16 + 4.47i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (8.42 + 2.73i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.88753781086447420701439775245, −10.50639634622107774757645511869, −9.768891227041784766949514277417, −8.315350733697980635283375912962, −7.39322049951702893843374504154, −6.81108536671810736494322676323, −5.41513356990203817475419675385, −4.47845220995514335667623601175, −3.39269652150154048344834074429, −1.41471885095522751821427263707,
2.24787356005077656924115650226, 2.80114168250959235066036832150, 4.18644551030163123263051204546, 5.41974301090736738399078334392, 6.61595137107513362568636493511, 7.86768326351568074952223064449, 8.678411546986845845122078811881, 9.320023949603302701683346145285, 10.84226022332848189271850419811, 11.52614889452672521733485048501