| L(s) = 1 | − 2-s + 4-s − 8-s − 11-s − 4·13-s + 16-s + 4·17-s − 19-s + 22-s + 5·23-s + 4·26-s − 3·29-s − 5·31-s − 32-s − 4·34-s + 6·37-s + 38-s + 2·41-s − 4·43-s − 44-s − 5·46-s − 7·49-s − 4·52-s + 9·53-s + 3·58-s − 11·61-s + 5·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.301·11-s − 1.10·13-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.213·22-s + 1.04·23-s + 0.784·26-s − 0.557·29-s − 0.898·31-s − 0.176·32-s − 0.685·34-s + 0.986·37-s + 0.162·38-s + 0.312·41-s − 0.609·43-s − 0.150·44-s − 0.737·46-s − 49-s − 0.554·52-s + 1.23·53-s + 0.393·58-s − 1.40·61-s + 0.635·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.55014177587103111147052823456, −6.97023654954076281581871769579, −6.15144262904064091895506478186, −5.36371980725703358668387034290, −4.79492214677890238168714175791, −3.71715664640840112916031364668, −2.91317736573262974873414593450, −2.16138267270637622702864879925, −1.13954660723492745662196184427, 0,
1.13954660723492745662196184427, 2.16138267270637622702864879925, 2.91317736573262974873414593450, 3.71715664640840112916031364668, 4.79492214677890238168714175791, 5.36371980725703358668387034290, 6.15144262904064091895506478186, 6.97023654954076281581871769579, 7.55014177587103111147052823456
