L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 2·13-s + 14-s + 16-s − 20-s + 25-s − 2·26-s + 28-s − 6·29-s − 8·31-s + 32-s − 35-s + 4·37-s − 40-s − 6·41-s + 2·43-s − 6·47-s + 49-s + 50-s − 2·52-s − 6·53-s + 56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.169·35-s + 0.657·37-s − 0.158·40-s − 0.937·41-s + 0.304·43-s − 0.875·47-s + 1/7·49-s + 0.141·50-s − 0.277·52-s − 0.824·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.052203875\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.052203875\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−12.96543014183252, −12.54271524477858, −12.02428488001543, −11.56241079822403, −11.22677459302050, −10.74180751953583, −10.32215582627263, −9.590379941484310, −9.287915983335001, −8.675834341583134, −8.075126694045329, −7.541550646653955, −7.415051344526514, −6.639175000142097, −6.267213150314588, −5.548780503396564, −5.184660265937405, −4.682144326954850, −4.170807924484809, −3.562001126930553, −3.216154094099329, −2.437229636914841, −1.894899040670876, −1.338754601024280, −0.3359590102481302,
0.3359590102481302, 1.338754601024280, 1.894899040670876, 2.437229636914841, 3.216154094099329, 3.562001126930553, 4.170807924484809, 4.682144326954850, 5.184660265937405, 5.548780503396564, 6.267213150314588, 6.639175000142097, 7.415051344526514, 7.541550646653955, 8.075126694045329, 8.675834341583134, 9.287915983335001, 9.590379941484310, 10.32215582627263, 10.74180751953583, 11.22677459302050, 11.56241079822403, 12.02428488001543, 12.54271524477858, 12.96543014183252