L(s) = 1 | + (−0.951 − 0.309i)5-s + (0.587 + 0.809i)9-s + (−0.896 − 0.142i)13-s + (−1.76 + 0.896i)17-s + (0.809 + 0.587i)25-s + (−0.5 − 1.53i)29-s + (−0.309 + 1.95i)37-s + (−1.53 + 1.11i)41-s + (−0.309 − 0.951i)45-s − i·49-s + (−0.809 + 1.58i)53-s + (0.363 − 0.5i)61-s + (0.809 + 0.412i)65-s + (−0.896 + 0.142i)73-s + (−0.309 + 0.951i)81-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)5-s + (0.587 + 0.809i)9-s + (−0.896 − 0.142i)13-s + (−1.76 + 0.896i)17-s + (0.809 + 0.587i)25-s + (−0.5 − 1.53i)29-s + (−0.309 + 1.95i)37-s + (−1.53 + 1.11i)41-s + (−0.309 − 0.951i)45-s − i·49-s + (−0.809 + 1.58i)53-s + (0.363 − 0.5i)61-s + (0.809 + 0.412i)65-s + (−0.896 + 0.142i)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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3879973948\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3879973948\) |
\(L(1)\) |
|
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 | \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \) |
| 17 | \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 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 | \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \) |
| 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 | \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \) |
| 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 | \( 1 + (0.896 - 0.142i)T + (0.951 - 0.309i)T^{2} \) |
| 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 | \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−8.991085503392421638962172814115, −8.225156376007592145573192611396, −7.79660021561954848219919316235, −6.94972738588872957739191273066, −6.28311336000867066295811535383, −4.97226520101109751194688508896, −4.59945269496535077803201736256, −3.80498330943995031609068761867, −2.63336358978007568798857050090, −1.63400017221592345329292108300,
0.22394509308416144064359145954, 1.93773845971010464790778987993, 3.02729866320802390578541883153, 3.88716921322603659933419767258, 4.57468277406314491065732305745, 5.38515286828146472738462455871, 6.77418861180630627128800672426, 6.92651089877222477871138613692, 7.62528978160584760117439320474, 8.748392789636756702991710484177