L(s) = 1 | + (−1.61 − 0.618i)3-s − 5-s + (−2.61 + 0.381i)7-s + (2.23 + 2.00i)9-s − 4.47i·11-s + 1.23i·13-s + (1.61 + 0.618i)15-s − 5.23·17-s + 8.47i·19-s + (4.47 + 1.00i)21-s + 4i·23-s + 25-s + (−2.38 − 4.61i)27-s − 7.70i·29-s − 2.76i·31-s + ⋯ |
L(s) = 1 | + (−0.934 − 0.356i)3-s − 0.447·5-s + (−0.989 + 0.144i)7-s + (0.745 + 0.666i)9-s − 1.34i·11-s + 0.342i·13-s + (0.417 + 0.159i)15-s − 1.26·17-s + 1.94i·19-s + (0.975 + 0.218i)21-s + 0.834i·23-s + 0.200·25-s + (−0.458 − 0.888i)27-s − 1.43i·29-s − 0.496i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7256863884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7256863884\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 + 0.618i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.61 - 0.381i)T \) |
good | 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 - 1.23iT - 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 8.47iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 7.70iT - 29T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 - 0.763T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 - 4.94T + 43T^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 - 0.472iT - 53T^{2} \) |
| 59 | \( 1 - 4.47T + 59T^{2} \) |
| 61 | \( 1 + 7.23iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 7.23iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 0.763iT - 97T^{2} \) |
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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
−9.462539222980154079589368042974, −8.380194129413061115993578415359, −7.73850230072972931309969015451, −6.69909144246749563671590176725, −6.10654048442219512628034356132, −5.54702520194793154193023487394, −4.22393615570992350387789236651, −3.52823915824955971478033856130, −2.14114945594425393114103069121, −0.60884747366574297473818730934,
0.58678745570981596150205855784, 2.39748030781449297594046047643, 3.58697614332718477325193177654, 4.62978828902664395454577419583, 4.99654695194061838572790554390, 6.43840980167514747513930066702, 6.78604218205584190752907784851, 7.48359838593633285781537883827, 8.954156046806161441729402664219, 9.311783856735645338077831975628