| L(s) = 1 | + 2·3-s − 2·5-s + 9-s − 11-s + 2·13-s − 4·15-s + 2·19-s − 25-s − 4·27-s + 6·29-s − 4·31-s − 2·33-s + 2·37-s + 4·39-s + 8·41-s − 12·43-s − 2·45-s − 12·47-s − 2·53-s + 2·55-s + 4·57-s − 10·59-s − 10·61-s − 4·65-s + 12·67-s − 4·71-s + 12·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.03·15-s + 0.458·19-s − 1/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.328·37-s + 0.640·39-s + 1.24·41-s − 1.82·43-s − 0.298·45-s − 1.75·47-s − 0.274·53-s + 0.269·55-s + 0.529·57-s − 1.30·59-s − 1.28·61-s − 0.496·65-s + 1.46·67-s − 0.474·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.82619487448678845685486744928, −6.88420472227818731407746511896, −6.17954231644219072510468276327, −5.21957456919730061256036186751, −4.46334868900367902524257023299, −3.60662757255630072193242606933, −3.23400828637191869712477583111, −2.37777618848739236526734684860, −1.37085742552661853239671743807, 0,
1.37085742552661853239671743807, 2.37777618848739236526734684860, 3.23400828637191869712477583111, 3.60662757255630072193242606933, 4.46334868900367902524257023299, 5.21957456919730061256036186751, 6.17954231644219072510468276327, 6.88420472227818731407746511896, 7.82619487448678845685486744928
