| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s + 13-s − 2·14-s + 16-s + 2·19-s − 20-s + 25-s + 26-s − 2·28-s − 6·29-s + 4·31-s + 32-s + 2·35-s + 10·37-s + 2·38-s − 40-s − 6·41-s − 10·43-s − 3·49-s + 50-s + 52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.458·19-s − 0.223·20-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.338·35-s + 1.64·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s − 1.52·43-s − 3/7·49-s + 0.141·50-s + 0.138·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.412043463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.412043463\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.69931865819286, −12.02952662773932, −11.78259091695917, −11.35567604789370, −10.90916489729899, −10.26215405193461, −9.979419538946435, −9.415733143372559, −8.968446827572050, −8.378400536256694, −7.819784274223955, −7.556660264431077, −6.792545598001337, −6.586513940857159, −6.041263720421430, −5.503703839074808, −5.054599726420777, −4.431160382654500, −4.011406912097037, −3.407335253514643, −3.103905016388475, −2.517615334451384, −1.797591068907406, −1.163082086102990, −0.3674579018617485,
0.3674579018617485, 1.163082086102990, 1.797591068907406, 2.517615334451384, 3.103905016388475, 3.407335253514643, 4.011406912097037, 4.431160382654500, 5.054599726420777, 5.503703839074808, 6.041263720421430, 6.586513940857159, 6.792545598001337, 7.556660264431077, 7.819784274223955, 8.378400536256694, 8.968446827572050, 9.415733143372559, 9.979419538946435, 10.26215405193461, 10.90916489729899, 11.35567604789370, 11.78259091695917, 12.02952662773932, 12.69931865819286
