| L(s) = 1 | − 3·5-s + 11-s + 2·13-s − 3·17-s − 2·19-s + 3·23-s + 4·25-s + 2·31-s − 8·37-s − 9·41-s − 4·43-s − 3·47-s + 6·53-s − 3·55-s + 6·59-s + 5·61-s − 6·65-s + 11·67-s − 2·73-s + 13·79-s + 9·83-s + 9·85-s + 12·89-s + 6·95-s − 5·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.301·11-s + 0.554·13-s − 0.727·17-s − 0.458·19-s + 0.625·23-s + 4/5·25-s + 0.359·31-s − 1.31·37-s − 1.40·41-s − 0.609·43-s − 0.437·47-s + 0.824·53-s − 0.404·55-s + 0.781·59-s + 0.640·61-s − 0.744·65-s + 1.34·67-s − 0.234·73-s + 1.46·79-s + 0.987·83-s + 0.976·85-s + 1.27·89-s + 0.615·95-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659802481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659802481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.53226045347659, −12.15306103151056, −11.71418196134385, −11.27761689839798, −11.04376438806871, −10.37246688542971, −10.05812284395554, −9.371770693320314, −8.683993834892433, −8.582341950932325, −8.185260137993021, −7.465743478540468, −7.137659700743727, −6.550127483963965, −6.345921115830789, −5.453796903227428, −4.955585454422148, −4.561168650858114, −3.924999468619954, −3.464421054550094, −3.259622204646663, −2.222856778181213, −1.864342086455771, −0.8933380013223365, −0.4247662394780629,
0.4247662394780629, 0.8933380013223365, 1.864342086455771, 2.222856778181213, 3.259622204646663, 3.464421054550094, 3.924999468619954, 4.561168650858114, 4.955585454422148, 5.453796903227428, 6.345921115830789, 6.550127483963965, 7.137659700743727, 7.465743478540468, 8.185260137993021, 8.582341950932325, 8.683993834892433, 9.371770693320314, 10.05812284395554, 10.37246688542971, 11.04376438806871, 11.27761689839798, 11.71418196134385, 12.15306103151056, 12.53226045347659
