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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.542433431\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542433431\) |
\(L(1)\) |
|
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
−8.908909039145596992389405141287, −8.175052354192274336983341805327, −7.02921019544704755726196527546, −6.45162545666089834369227336558, −5.90666254266253228148474384820, −4.79813026587707380752505612964, −4.23757910156487224304790595559, −3.06768138586088866772665989695, −2.10219786304855690986400064380, −1.00176787655974913117146060524,
1.60842696854187760692360529991, 2.18147198244407397291609807311, 3.38283107348994227472426037544, 4.35772636762039641551769448510, 5.14552965277944324194331920234, 5.99592970667031329937516755013, 6.72075262256837512559005734315, 7.23112017668530620485462132281, 8.520961612356084321238471350587, 8.727146995308200472057641120842