| L(s) = 1 | − 2.20·3-s + (−1.71 − 1.43i)5-s + 0.642i·7-s + 1.85·9-s + 2.87·11-s − 0.405i·13-s + (3.78 + 3.15i)15-s − 4.93·17-s + 2.74·19-s − 1.41i·21-s + (4.71 + 0.896i)23-s + (0.900 + 4.91i)25-s + 2.52·27-s − 1.62·29-s + 4.55i·31-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + (−0.768 − 0.640i)5-s + 0.242i·7-s + 0.618·9-s + 0.865·11-s − 0.112i·13-s + (0.977 + 0.814i)15-s − 1.19·17-s + 0.629·19-s − 0.308i·21-s + (0.982 + 0.186i)23-s + (0.180 + 0.983i)25-s + 0.485·27-s − 0.302·29-s + 0.818i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5360826200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5360826200\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.71 + 1.43i)T \) |
| 23 | \( 1 + (-4.71 - 0.896i)T \) |
| good | 3 | \( 1 + 2.20T + 3T^{2} \) |
| 7 | \( 1 - 0.642iT - 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 + 0.405iT - 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 - 4.55iT - 31T^{2} \) |
| 37 | \( 1 - 2.00T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 + 3.29iT - 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 + 0.263T + 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 9.19iT - 61T^{2} \) |
| 67 | \( 1 - 2.18iT - 67T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 4.43iT - 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 + 9.14iT - 83T^{2} \) |
| 89 | \( 1 + 14.0iT - 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.940364058151443615715945668144, −8.368194800357736283524521620234, −7.16168417690318043293619792207, −6.67650705131915713368581993429, −5.65937842112484572640754059055, −4.98879758564558186651989413527, −4.27484227678064000714671661415, −3.19718227271936603208046899473, −1.49431842685269647804584352950, −0.30753755167874450847533999261,
1.00686088898891653982886726054, 2.66806427039613093219504926799, 3.86895892747397367156662675076, 4.55447039993761673521763052161, 5.51558960574135744789398444082, 6.51638843848176304562615828504, 6.83627817112874454430215081956, 7.71324384219903935084862572843, 8.748350306031905532524145550917, 9.545081549443667870883568481758
