L(s) = 1 | + 5-s + 2.82·7-s + 5.65·11-s + 2·13-s − 2·17-s + 2.82·23-s + 25-s + 6·29-s − 5.65·31-s + 2.82·35-s + 10·37-s − 2·41-s − 8.48·43-s − 2.82·47-s + 1.00·49-s + 6·53-s + 5.65·55-s − 11.3·59-s + 2·61-s + 2·65-s − 2.82·67-s + 5.65·71-s − 6·73-s + 16.0·77-s − 11.3·79-s − 2.82·83-s − 2·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.06·7-s + 1.70·11-s + 0.554·13-s − 0.485·17-s + 0.589·23-s + 0.200·25-s + 1.11·29-s − 1.01·31-s + 0.478·35-s + 1.64·37-s − 0.312·41-s − 1.29·43-s − 0.412·47-s + 0.142·49-s + 0.824·53-s + 0.762·55-s − 1.47·59-s + 0.256·61-s + 0.248·65-s − 0.345·67-s + 0.671·71-s − 0.702·73-s + 1.82·77-s − 1.27·79-s − 0.310·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.708241077\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.708241077\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2.82T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.764855317141859059948771454605, −8.198067088899452712774806888709, −7.15369022962484438432350467669, −6.50212684857018536660123104448, −5.78383733171085365877461415670, −4.76776905303053747352123289394, −4.17791477177843098677326972084, −3.12260192251420575037882017513, −1.84584020790229544377387987317, −1.15213211861402453714121810002,
1.15213211861402453714121810002, 1.84584020790229544377387987317, 3.12260192251420575037882017513, 4.17791477177843098677326972084, 4.76776905303053747352123289394, 5.78383733171085365877461415670, 6.50212684857018536660123104448, 7.15369022962484438432350467669, 8.198067088899452712774806888709, 8.764855317141859059948771454605