| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 13-s + 14-s + 16-s − 3·17-s + 19-s + 3·23-s + 26-s + 28-s + 3·29-s + 2·31-s + 32-s − 3·34-s + 10·37-s + 38-s − 6·41-s − 2·43-s + 3·46-s − 6·49-s + 52-s + 3·53-s + 56-s + 3·58-s − 3·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.625·23-s + 0.196·26-s + 0.188·28-s + 0.557·29-s + 0.359·31-s + 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.162·38-s − 0.937·41-s − 0.304·43-s + 0.442·46-s − 6/7·49-s + 0.138·52-s + 0.412·53-s + 0.133·56-s + 0.393·58-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.673264451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.673264451\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.80897041789104760264366694659, −6.83479656039264157250441484643, −6.48437227109576585924904451976, −5.61104712058905754691187680203, −4.91653560927365896279618152228, −4.36983883097323894026662942215, −3.52804456552046371419725536875, −2.74021131009157130656689341376, −1.91292203307911529502364772277, −0.857166394202570666551052750498,
0.857166394202570666551052750498, 1.91292203307911529502364772277, 2.74021131009157130656689341376, 3.52804456552046371419725536875, 4.36983883097323894026662942215, 4.91653560927365896279618152228, 5.61104712058905754691187680203, 6.48437227109576585924904451976, 6.83479656039264157250441484643, 7.80897041789104760264366694659
