L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 6·11-s − 2·13-s + 14-s + 16-s + 4·17-s − 20-s + 6·22-s + 4·23-s + 25-s + 2·26-s − 28-s + 6·29-s − 6·31-s − 32-s − 4·34-s + 35-s − 4·37-s + 40-s + 6·41-s − 6·44-s − 4·46-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 − 1.80·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 1.27·22-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s + 0.169·35-s − 0.657·37-s + 0.158·40-s + 0.937·41-s − 0.904·44-s − 0.589·46-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.89395133348985, −12.66199250369652, −12.37847926562389, −11.68435268312502, −11.23205410943639, −10.73921961172437, −10.30068238675260, −10.03348818526098, −9.463237413170407, −8.924060701332632, −8.442603619423127, −7.938314615235117, −7.533147700049064, −7.210744775726492, −6.691692347379341, −5.844287049509866, −5.638558807466807, −4.853694454162372, −4.635235731226950, −3.649929280492077, −3.055236396607461, −2.840279557446330, −2.128957485015117, −1.379707234507697, −0.5906701179000906, 0,
0.5906701179000906, 1.379707234507697, 2.128957485015117, 2.840279557446330, 3.055236396607461, 3.649929280492077, 4.635235731226950, 4.853694454162372, 5.638558807466807, 5.844287049509866, 6.691692347379341, 7.210744775726492, 7.533147700049064, 7.938314615235117, 8.442603619423127, 8.924060701332632, 9.463237413170407, 10.03348818526098, 10.30068238675260, 10.73921961172437, 11.23205410943639, 11.68435268312502, 12.37847926562389, 12.66199250369652, 12.89395133348985