L(s) = 1 | + 0.580i·3-s + (−0.209 − 2.22i)5-s + 0.315i·7-s + 2.66·9-s + 4.34·11-s + 4.58i·13-s + (1.29 − 0.121i)15-s − 0.917i·17-s + 2.76·19-s − 0.182·21-s + i·23-s + (−4.91 + 0.933i)25-s + 3.28i·27-s − 7.03·29-s + 0.867·31-s + ⋯ |
L(s) = 1 | + 0.335i·3-s + (−0.0938 − 0.995i)5-s + 0.119i·7-s + 0.887·9-s + 1.31·11-s + 1.27i·13-s + (0.333 − 0.0314i)15-s − 0.222i·17-s + 0.634·19-s − 0.0399·21-s + 0.208i·23-s + (−0.982 + 0.186i)25-s + 0.632i·27-s − 1.30·29-s + 0.155·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.019619012\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.019619012\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.209 + 2.22i)T \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 0.580iT - 3T^{2} \) |
| 7 | \( 1 - 0.315iT - 7T^{2} \) |
| 11 | \( 1 - 4.34T + 11T^{2} \) |
| 13 | \( 1 - 4.58iT - 13T^{2} \) |
| 17 | \( 1 + 0.917iT - 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 + 7.03T + 29T^{2} \) |
| 31 | \( 1 - 0.867T + 31T^{2} \) |
| 37 | \( 1 + 4.68iT - 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 + 9.08iT - 43T^{2} \) |
| 47 | \( 1 - 8.24iT - 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 1.68iT - 67T^{2} \) |
| 71 | \( 1 - 5.36T + 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 - 7.39T + 79T^{2} \) |
| 83 | \( 1 - 15.3iT - 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−9.343660267973542050879925598470, −8.798039965566747127425499513874, −7.59364738678647366028615812988, −7.00094698525432743589670652647, −5.99164147330214635408141334956, −5.08544675293221395898432592117, −4.14421309768067775905920718015, −3.81991768666813546412031111495, −2.01478749951440108450856737493, −1.09074619162606516455815478407,
0.994225928135952564754297496753, 2.21913288680668041513862910580, 3.42548824719958414197157570818, 4.01435628543768837725678323114, 5.28988294614963044482685625726, 6.26765752684341133907792527717, 6.87897613103876338686662826336, 7.56021032597850412147044517930, 8.255977654527160604986001956067, 9.427488625035361465321521250331