L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s + 2·13-s + 14-s + 16-s + 2·17-s + 19-s − 2·20-s − 4·23-s − 25-s − 2·26-s − 28-s + 2·29-s − 32-s − 2·34-s + 2·35-s − 2·37-s − 38-s + 2·40-s + 6·41-s + 4·43-s + 4·46-s + 49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.447·20-s − 0.834·23-s − 1/5·25-s − 0.392·26-s − 0.188·28-s + 0.371·29-s − 0.176·32-s − 0.342·34-s + 0.338·35-s − 0.328·37-s − 0.162·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + 0.589·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−8.582949550428574214691197938914, −7.77303490791181199394105951616, −7.35939035562692224214671398184, −6.31926009244706325425517219737, −5.67423501212729589177501898203, −4.38105634478578668336094152391, −3.61706432739133420518803858039, −2.68579729138873371401615998900, −1.31643594305533919457649487988, 0,
1.31643594305533919457649487988, 2.68579729138873371401615998900, 3.61706432739133420518803858039, 4.38105634478578668336094152391, 5.67423501212729589177501898203, 6.31926009244706325425517219737, 7.35939035562692224214671398184, 7.77303490791181199394105951616, 8.582949550428574214691197938914