L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s − 7-s + 9-s + 2·10-s + 11-s + 2·12-s − 2·13-s + 2·14-s − 15-s − 4·16-s + 17-s − 2·18-s − 7·19-s − 2·20-s − 21-s − 2·22-s + 5·23-s + 25-s + 4·26-s + 27-s − 2·28-s + 3·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s − 0.258·15-s − 16-s + 0.242·17-s − 0.471·18-s − 1.60·19-s − 0.447·20-s − 0.218·21-s − 0.426·22-s + 1.04·23-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s + 0.557·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.232883867315003728235264073130, −8.624798047867196590682284722049, −8.038415560752709454825674143240, −7.08341510627234054758361469624, −6.59116630616645929033052834068, −4.98917827881150120204901528781, −3.95223421515006378458787650566, −2.76005954812171290376319808489, −1.57102768387950557019329678346, 0,
1.57102768387950557019329678346, 2.76005954812171290376319808489, 3.95223421515006378458787650566, 4.98917827881150120204901528781, 6.59116630616645929033052834068, 7.08341510627234054758361469624, 8.038415560752709454825674143240, 8.624798047867196590682284722049, 9.232883867315003728235264073130