L(s) = 1 | + (0.951 − 0.309i)5-s + (−0.587 + 0.809i)9-s + (0.278 + 1.76i)13-s + (0.142 − 0.278i)17-s + (0.809 − 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.309 + 0.0489i)37-s + (1.53 + 1.11i)41-s + (−0.309 + 0.951i)45-s − i·49-s + (−0.809 + 0.412i)53-s + (−0.363 − 0.5i)61-s + (0.809 + 1.58i)65-s + (0.278 − 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)5-s + (−0.587 + 0.809i)9-s + (0.278 + 1.76i)13-s + (0.142 − 0.278i)17-s + (0.809 − 0.587i)25-s + (−0.5 + 1.53i)29-s + (−0.309 + 0.0489i)37-s + (1.53 + 1.11i)41-s + (−0.309 + 0.951i)45-s − i·49-s + (−0.809 + 0.412i)53-s + (−0.363 − 0.5i)61-s + (0.809 + 1.58i)65-s + (0.278 − 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.411354577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.411354577\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.951 + 0.309i)T \) |
good | 3 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.278 - 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.76 + 0.896i)T + (0.587 - 0.809i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.138983840347317819625382775696, −8.320416777627944740739642384098, −7.41274747817578446035947141491, −6.58691306830030616668050220732, −5.94555987334737690863189070481, −5.06645575810980661688153033036, −4.51853735510722989318896202187, −3.29300853141968039400977298610, −2.23382739782826012951355945531, −1.51687531081595597247784871128,
0.896026578594829711902654803981, 2.31117296637407786983136546815, 3.07811874828624651049161962562, 3.90517311460152059876187266516, 5.17428467319671046747660080506, 5.94671991008212414041771138822, 6.14629207564981743680157257263, 7.31314743931621969018464562745, 8.017020126232995486257334859814, 8.836835524597299596222750400987