L(s) = 1 | + (−0.587 − 0.809i)9-s + (−0.278 + 1.76i)13-s + (−1.76 + 0.896i)17-s + (−1.11 + 0.363i)29-s + (−0.309 − 0.0489i)37-s + (−1.53 + 1.11i)41-s − i·49-s + (0.809 + 0.412i)53-s + (1.53 + 1.11i)61-s + (−0.896 + 0.142i)73-s + (−0.309 + 0.951i)81-s + (−0.690 + 0.951i)89-s + (−0.142 + 0.278i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)9-s + (−0.278 + 1.76i)13-s + (−1.76 + 0.896i)17-s + (−1.11 + 0.363i)29-s + (−0.309 − 0.0489i)37-s + (−1.53 + 1.11i)41-s − i·49-s + (0.809 + 0.412i)53-s + (1.53 + 1.11i)61-s + (−0.896 + 0.142i)73-s + (−0.309 + 0.951i)81-s + (−0.690 + 0.951i)89-s + (−0.142 + 0.278i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5454210538\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5454210538\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
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 | \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 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 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 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
−8.780312714046468729201109606237, −8.486165683131870647605389181444, −7.18106959787945224830752980225, −6.70600394552555355152902566726, −6.11201174933033780732850014524, −5.12433608789207800505249670657, −4.21569492320499818761247706300, −3.67468320598989029457680935535, −2.44437199663304278581876360370, −1.63611253974029431205156335384,
0.27872918691562738619872904327, 2.04503085101582213803535825315, 2.74425083124120998568537263717, 3.66515482966975001347982213215, 4.80685494652666614713014052584, 5.29817380628038188944813623945, 6.04787870666408704959346822591, 7.06405306828222902180341786055, 7.60365246983964020145500405557, 8.445785664905282110209859723514