| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 6·11-s − 13-s − 14-s + 16-s − 3·17-s + 2·19-s + 6·22-s + 26-s + 28-s − 6·29-s − 4·31-s − 32-s + 3·34-s + 7·37-s − 2·38-s + 43-s − 6·44-s + 3·47-s − 6·49-s − 52-s − 56-s + 6·58-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.80·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 1.27·22-s + 0.196·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s + 1.15·37-s − 0.324·38-s + 0.152·43-s − 0.904·44-s + 0.437·47-s − 6/7·49-s − 0.138·52-s − 0.133·56-s + 0.787·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8983626244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8983626244\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.84938842527182748510711115216, −7.72883043526770917154309219588, −6.91074989282661788883788110479, −5.94367988122443797135544909606, −5.29558150515983084216968353717, −4.60443486063485603504528049329, −3.47943454185054069065437909300, −2.54122906163647977924246347543, −1.92671816756358203379527756743, −0.53271731854877995717434247878,
0.53271731854877995717434247878, 1.92671816756358203379527756743, 2.54122906163647977924246347543, 3.47943454185054069065437909300, 4.60443486063485603504528049329, 5.29558150515983084216968353717, 5.94367988122443797135544909606, 6.91074989282661788883788110479, 7.72883043526770917154309219588, 7.84938842527182748510711115216
