L(s) = 1 | − 3-s + 4·5-s − 2·7-s + 9-s − 3·11-s − 4·15-s + 3·17-s + 2·21-s + 6·23-s + 11·25-s − 27-s + 5·29-s + 7·31-s + 3·33-s − 8·35-s + 7·37-s − 41-s + 43-s + 4·45-s + 3·47-s − 3·49-s − 3·51-s − 6·53-s − 12·55-s − 3·61-s − 2·63-s − 2·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.755·7-s + 1/3·9-s − 0.904·11-s − 1.03·15-s + 0.727·17-s + 0.436·21-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.928·29-s + 1.25·31-s + 0.522·33-s − 1.35·35-s + 1.15·37-s − 0.156·41-s + 0.152·43-s + 0.596·45-s + 0.437·47-s − 3/7·49-s − 0.420·51-s − 0.824·53-s − 1.61·55-s − 0.384·61-s − 0.251·63-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.307918945\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.307918945\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−12.73172156786137, −12.26445650088573, −11.72688594482157, −11.09205050814083, −10.59776573936211, −10.35673055396608, −9.807997303752640, −9.610715203433637, −9.135659684235512, −8.527192463077840, −7.993488565690042, −7.420693665168302, −6.790378798311656, −6.410431551713842, −6.134005088514207, −5.472015569659149, −5.274082349388903, −4.697558846453762, −4.178307001543390, −3.093542621451908, −2.917268334872720, −2.456630262431532, −1.634137824989851, −1.111991199051465, −0.5320393264007873,
0.5320393264007873, 1.111991199051465, 1.634137824989851, 2.456630262431532, 2.917268334872720, 3.093542621451908, 4.178307001543390, 4.697558846453762, 5.274082349388903, 5.472015569659149, 6.134005088514207, 6.410431551713842, 6.790378798311656, 7.420693665168302, 7.993488565690042, 8.527192463077840, 9.135659684235512, 9.610715203433637, 9.807997303752640, 10.35673055396608, 10.59776573936211, 11.09205050814083, 11.72688594482157, 12.26445650088573, 12.73172156786137