| L(s) = 1 | − 2-s + 4-s − 8-s + 13-s + 16-s + 3·17-s + 19-s − 26-s − 6·29-s − 2·31-s − 32-s − 3·34-s + 10·37-s − 38-s + 3·41-s + 43-s + 12·47-s + 52-s + 6·53-s + 6·58-s + 6·59-s − 14·61-s + 2·62-s + 64-s + 10·67-s + 3·68-s + 3·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.229·19-s − 0.196·26-s − 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.514·34-s + 1.64·37-s − 0.162·38-s + 0.468·41-s + 0.152·43-s + 1.75·47-s + 0.138·52-s + 0.824·53-s + 0.787·58-s + 0.781·59-s − 1.79·61-s + 0.254·62-s + 1/8·64-s + 1.22·67-s + 0.363·68-s + 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.800989313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800989313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−12.59908862758496, −12.26926128222961, −11.71010538560030, −11.24793434682899, −10.91707711371693, −10.40804912186360, −9.922525452837445, −9.395101009694037, −9.200700895434530, −8.540682554489413, −8.100284461861383, −7.613851311388945, −7.226416159873039, −6.804152689564234, −5.976477236499305, −5.773925875509099, −5.323950235447380, −4.461488472280050, −4.020267409533149, −3.486216856650417, −2.765991776335980, −2.405812179593703, −1.603231110849663, −1.084223208263979, −0.4479635672533044,
0.4479635672533044, 1.084223208263979, 1.603231110849663, 2.405812179593703, 2.765991776335980, 3.486216856650417, 4.020267409533149, 4.461488472280050, 5.323950235447380, 5.773925875509099, 5.976477236499305, 6.804152689564234, 7.226416159873039, 7.613851311388945, 8.100284461861383, 8.540682554489413, 9.200700895434530, 9.395101009694037, 9.922525452837445, 10.40804912186360, 10.91707711371693, 11.24793434682899, 11.71010538560030, 12.26926128222961, 12.59908862758496
