L(s) = 1 | + (0.707 + 0.707i)3-s + (1.98 + 1.03i)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 + (0.887 + 0.461i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.844961029\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.844961029\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.98 - 1.03i)T \) |
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
−9.164058316279048129514776366319, −8.460491968140777724020094263995, −7.903248319077171416633807038289, −6.93258668928515688299100299020, −5.95318369423711529408405304750, −5.08290079456761835320796925409, −4.58112792354302579804455104977, −3.09806192959608393220870188254, −2.48807007274003333903086402363, −1.24509339572994510949771223826,
1.28633013153983547917468989265, 1.95393086074793446112407063270, 2.98022739208957945853566252941, 4.51209973018711212518105181116, 5.01046602826570197415528536787, 5.88065207530725199940414337712, 6.86239379408615064751972187871, 7.81965422368097116027575419534, 8.284672501183672424686877498963, 8.969512466325923515910324798570