L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 + 1.73i)11-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)15-s − 17-s + 0.999·21-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)29-s + 1.99·33-s + 0.999·35-s + (0.499 − 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1 + 1.73i)11-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)15-s − 17-s + 0.999·21-s + (−0.499 + 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)29-s + 1.99·33-s + 0.999·35-s + (0.499 − 0.866i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4133396438\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4133396438\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 11 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.960404819097634586826001883482, −9.161810921855768919372592667284, −8.369207584681169553210032516886, −7.60352544086828354206564663855, −6.78640693361545804290467268543, −5.97098949285781605426479310695, −4.95586912477766349241103116159, −4.35778895576497876849218120891, −2.60576088608570906903360334815, −1.71354733364478025856655651242,
0.36539991274136573843907421278, 3.05916145408168353942793336503, 3.38628760539103877969484622366, 4.45849541859484611406644373857, 5.58986539697941753211106472356, 6.29577245621593577565232758370, 7.12834963442663261870102134463, 8.209836644865922031817893005036, 8.813447934972045669805243853457, 10.17571133057321982192217242226