L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.03 − 1.98i)5-s − 3.91·7-s − 1.00i·9-s + (−2.93 + 2.93i)11-s + (−0.732 + 0.732i)13-s + (0.674 + 2.13i)15-s − 2.89i·17-s + (1.67 + 1.67i)19-s + (2.77 − 2.77i)21-s − 1.73·23-s + (−2.87 − 4.09i)25-s + (0.707 + 0.707i)27-s + (4.99 + 4.99i)29-s + 10.8·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.461 − 0.887i)5-s − 1.48·7-s − 0.333i·9-s + (−0.884 + 0.884i)11-s + (−0.203 + 0.203i)13-s + (0.174 + 0.550i)15-s − 0.701i·17-s + (0.384 + 0.384i)19-s + (0.604 − 0.604i)21-s − 0.361·23-s + (−0.574 − 0.818i)25-s + (0.136 + 0.136i)27-s + (0.926 + 0.926i)29-s + 1.94·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.121649854\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.121649854\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-1.03 + 1.98i)T \) |
good | 7 | \( 1 + 3.91T + 7T^{2} \) |
| 11 | \( 1 + (2.93 - 2.93i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.732 - 0.732i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.89iT - 17T^{2} \) |
| 19 | \( 1 + (-1.67 - 1.67i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 + (-4.99 - 4.99i)T + 29iT^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + (-6.41 - 6.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.00577iT - 41T^{2} \) |
| 43 | \( 1 + (-2.23 - 2.23i)T + 43iT^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + (-5.55 - 5.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.83 + 3.83i)T - 59iT^{2} \) |
| 61 | \( 1 + (9.30 + 9.30i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.85 + 3.85i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.15iT - 71T^{2} \) |
| 73 | \( 1 - 7.98T + 73T^{2} \) |
| 79 | \( 1 + 0.843T + 79T^{2} \) |
| 83 | \( 1 + (-5.20 + 5.20i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.40iT - 89T^{2} \) |
| 97 | \( 1 - 2.24iT - 97T^{2} \) |
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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
−9.467902139720975199892906903296, −8.609605421399211275574231424795, −7.65582528648929303764662890239, −6.64500546581562547388050767879, −6.07450870959553184931371743503, −5.03189981047754804515895658428, −4.59028307914704651848971439237, −3.32263895463691316335295123062, −2.36646446257167211718513806247, −0.74106377521638710592325901634,
0.66744686968484171720252043409, 2.55361254896048587342174559941, 2.93825320796162715325887954094, 4.13242686878794427173804453724, 5.52707300577951402586587656048, 6.14317184392113087552553060294, 6.56447915626923749802759808511, 7.52934082226499316423887674728, 8.248984312824203428558800452006, 9.376511476763441164770158362242