| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 2·11-s − 15-s + 4·21-s + 8·23-s + 25-s + 27-s − 2·29-s + 2·31-s − 2·33-s − 4·35-s + 8·37-s − 2·41-s + 4·43-s − 45-s + 9·49-s + 6·53-s + 2·55-s − 14·59-s + 14·61-s + 4·63-s − 4·67-s + 8·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 0.258·15-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.348·33-s − 0.676·35-s + 1.31·37-s − 0.312·41-s + 0.609·43-s − 0.149·45-s + 9/7·49-s + 0.824·53-s + 0.269·55-s − 1.82·59-s + 1.79·61-s + 0.503·63-s − 0.488·67-s + 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.408141160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408141160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.905958225250015730932642240476, −8.475697246379034923748800257665, −7.58903760041826735478913178216, −7.27919207522379862388962403738, −5.92704687742133064020277025243, −4.90827321544753146649955163789, −4.42748833343524287397809510890, −3.23500564434491538144898168837, −2.27720663910374506444794308813, −1.09505201742716397244922543371,
1.09505201742716397244922543371, 2.27720663910374506444794308813, 3.23500564434491538144898168837, 4.42748833343524287397809510890, 4.90827321544753146649955163789, 5.92704687742133064020277025243, 7.27919207522379862388962403738, 7.58903760041826735478913178216, 8.475697246379034923748800257665, 8.905958225250015730932642240476
