L(s) = 1 | + (−2.13 + 0.337i)3-s + 2.23·5-s + (−1.09 − 1.09i)7-s + (1.58 − 0.514i)9-s + (1.97 + 0.642i)11-s + (0.587 − 1.15i)13-s + (−4.77 + 0.755i)15-s + (−0.224 + 1.41i)17-s + (6.56 − 4.77i)19-s + (2.70 + 1.96i)21-s + (2.41 + 4.73i)23-s + 5.00·25-s + (2.56 − 1.30i)27-s + (3.65 − 5.02i)29-s + (3.63 + 4.99i)31-s + ⋯ |
L(s) = 1 | + (−1.23 + 0.195i)3-s + 0.999·5-s + (−0.413 − 0.413i)7-s + (0.528 − 0.171i)9-s + (0.596 + 0.193i)11-s + (0.163 − 0.319i)13-s + (−1.23 + 0.195i)15-s + (−0.0544 + 0.343i)17-s + (1.50 − 1.09i)19-s + (0.589 + 0.428i)21-s + (0.502 + 0.987i)23-s + 1.00·25-s + (0.494 − 0.251i)27-s + (0.678 − 0.933i)29-s + (0.651 + 0.897i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08564 - 0.0341176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08564 - 0.0341176i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
good | 3 | \( 1 + (2.13 - 0.337i)T + (2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (1.09 + 1.09i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.97 - 0.642i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.587 + 1.15i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (0.224 - 1.41i)T + (-16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-6.56 + 4.77i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 4.73i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-3.65 + 5.02i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.63 - 4.99i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.658 - 0.335i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-2.49 - 7.67i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (2.49 - 2.49i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.64 + 10.4i)T + (-44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (1.37 + 8.68i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.75 - 5.39i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.25 - 3.85i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (4.77 + 0.755i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (2.26 - 3.11i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (11.3 - 5.76i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (12.6 + 9.15i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.0651 + 0.411i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.613 - 0.199i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (17.7 - 2.81i)T + (92.2 - 29.9i)T^{2} \) |
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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
−11.38397999131398005468542171873, −10.22512830795043688867995369384, −9.822493509085418328333387992313, −8.712419881974727128486385570964, −7.14855102892366063534024773439, −6.39638668159706386584643503536, −5.52438338457967465981881451725, −4.69246810558496273281674861990, −3.07032819288458334513760624975, −1.10202006843746002913564316255,
1.21494058887090464628319425564, 2.96180829429824222042788746234, 4.71860332631535565277281602401, 5.79155428482984898956458066919, 6.21969896363172274474707403421, 7.21174584425289851893827954929, 8.765430158293419472493696838164, 9.567109342497923622495086397424, 10.45114448353650492000161015827, 11.33558869851283107668059528926