| L(s) = 1 | + 2·2-s + 5·3-s + 4·4-s + 10·6-s − 12·7-s + 8·8-s − 2·9-s + 22·11-s + 20·12-s − 19·13-s − 24·14-s + 16·16-s − 96·17-s − 4·18-s − 98·19-s − 60·21-s + 44·22-s − 23·23-s + 40·24-s − 38·26-s − 145·27-s − 48·28-s − 227·29-s − 285·31-s + 32·32-s + 110·33-s − 192·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.962·3-s + 1/2·4-s + 0.680·6-s − 0.647·7-s + 0.353·8-s − 0.0740·9-s + 0.603·11-s + 0.481·12-s − 0.405·13-s − 0.458·14-s + 1/4·16-s − 1.36·17-s − 0.0523·18-s − 1.18·19-s − 0.623·21-s + 0.426·22-s − 0.208·23-s + 0.340·24-s − 0.286·26-s − 1.03·27-s − 0.323·28-s − 1.45·29-s − 1.65·31-s + 0.176·32-s + 0.580·33-s − 0.968·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 98 T + p^{3} T^{2} \) |
| 29 | \( 1 + 227 T + p^{3} T^{2} \) |
| 31 | \( 1 + 285 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 271 T + p^{3} T^{2} \) |
| 43 | \( 1 - 100 T + p^{3} T^{2} \) |
| 47 | \( 1 - 285 T + p^{3} T^{2} \) |
| 53 | \( 1 + 18 T + p^{3} T^{2} \) |
| 59 | \( 1 + 352 T + p^{3} T^{2} \) |
| 61 | \( 1 + 478 T + p^{3} T^{2} \) |
| 67 | \( 1 + 330 T + p^{3} T^{2} \) |
| 71 | \( 1 - 835 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1127 T + p^{3} T^{2} \) |
| 79 | \( 1 - 322 T + p^{3} T^{2} \) |
| 83 | \( 1 + 572 T + p^{3} T^{2} \) |
| 89 | \( 1 + 504 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1712 T + p^{3} T^{2} \) |
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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
−9.274424966903598502395328163860, −8.160512675296373668773523147859, −7.31264039087180096262951875750, −6.43610080137948554697368691599, −5.69177130173455642917030403790, −4.33035628007977178360199372972, −3.76184515462180093107244420098, −2.66137586414364643666600437909, −1.95958605753214643009835872576, 0,
1.95958605753214643009835872576, 2.66137586414364643666600437909, 3.76184515462180093107244420098, 4.33035628007977178360199372972, 5.69177130173455642917030403790, 6.43610080137948554697368691599, 7.31264039087180096262951875750, 8.160512675296373668773523147859, 9.274424966903598502395328163860
