L(s) = 1 | − 2.41i·3-s − 0.828i·7-s − 2.82·9-s − 5.24·11-s − 5.65i·13-s + 0.171i·17-s + 1.58·19-s − 1.99·21-s − 4.82i·23-s − 0.414i·27-s − 8·29-s − 0.828·31-s + 12.6i·33-s + 7.65i·37-s − 13.6·39-s + ⋯ |
L(s) = 1 | − 1.39i·3-s − 0.313i·7-s − 0.942·9-s − 1.58·11-s − 1.56i·13-s + 0.0416i·17-s + 0.363·19-s − 0.436·21-s − 1.00i·23-s − 0.0797i·27-s − 1.48·29-s − 0.148·31-s + 2.20i·33-s + 1.25i·37-s − 2.18·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5946262451\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5946262451\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.41iT - 3T^{2} \) |
| 7 | \( 1 + 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 0.171iT - 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 - 7.65iT - 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 9.65iT - 47T^{2} \) |
| 53 | \( 1 - 7.65iT - 53T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 2.75iT - 67T^{2} \) |
| 71 | \( 1 + 9.65T + 71T^{2} \) |
| 73 | \( 1 - 5.82iT - 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 1.24iT - 83T^{2} \) |
| 89 | \( 1 + 6.17T + 89T^{2} \) |
| 97 | \( 1 - 17.3iT - 97T^{2} \) |
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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
−7.82540969291841712619769024398, −7.62377678578390705235283170449, −6.85353248474464675835538526969, −5.78644788559578376695634494450, −5.46276845800538196616417743550, −4.26989138065251087423972225246, −2.94637977471264276048510161505, −2.45468226332452055816394000756, −1.16807230865654579689269821329, −0.18764947194953940710212597448,
1.89836500422513399073781545863, 2.88772420141092971452159805080, 3.86700038999519217131011326156, 4.45511239226416405361604708017, 5.40503923442936961687093705651, 5.72188514815253540085561803480, 7.07853385104878671258178182076, 7.67081934355823400687103395202, 8.680269101792486623404565989164, 9.347177398737266649668279125149