| L(s) = 1 | − 2-s + 3-s + 4-s + 3·5-s − 6-s − 7-s − 8-s + 9-s − 3·10-s − 11-s + 12-s − 13-s + 14-s + 3·15-s + 16-s + 4·17-s − 18-s + 3·20-s − 21-s + 22-s − 4·23-s − 24-s + 4·25-s + 26-s + 27-s − 28-s + 9·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.670·20-s − 0.218·21-s + 0.213·22-s − 0.834·23-s − 0.204·24-s + 4/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 1.67·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.17908921636913, −13.58345463945964, −13.10022667077439, −12.74500381358421, −12.04087032858520, −11.74471501499651, −10.77952390603118, −10.47818742067409, −9.890799352767562, −9.638761325322758, −9.354722455516571, −8.429871830665203, −8.287796586934126, −7.631255600667307, −6.992300926272003, −6.514374787038548, −5.972510533835366, −5.505871795763100, −4.866614973027265, −4.133687164187703, −3.268983365680096, −2.841444203568487, −2.267088769336327, −1.630883708895884, −1.066338810561990, 0,
1.066338810561990, 1.630883708895884, 2.267088769336327, 2.841444203568487, 3.268983365680096, 4.133687164187703, 4.866614973027265, 5.505871795763100, 5.972510533835366, 6.514374787038548, 6.992300926272003, 7.631255600667307, 8.287796586934126, 8.429871830665203, 9.354722455516571, 9.638761325322758, 9.890799352767562, 10.47818742067409, 10.77952390603118, 11.74471501499651, 12.04087032858520, 12.74500381358421, 13.10022667077439, 13.58345463945964, 14.17908921636913
