| L(s) = 1 | + 3-s − 5-s − 2·9-s − 5·11-s + 13-s − 15-s − 3·17-s + 6·19-s + 6·23-s + 25-s − 5·27-s + 9·29-s − 5·33-s − 6·37-s + 39-s − 8·41-s + 6·43-s + 2·45-s + 3·47-s − 3·51-s + 12·53-s + 5·55-s + 6·57-s − 8·59-s − 4·61-s − 65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 1.50·11-s + 0.277·13-s − 0.258·15-s − 0.727·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s − 0.870·33-s − 0.986·37-s + 0.160·39-s − 1.24·41-s + 0.914·43-s + 0.298·45-s + 0.437·47-s − 0.420·51-s + 1.64·53-s + 0.674·55-s + 0.794·57-s − 1.04·59-s − 0.512·61-s − 0.124·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−16.13752399665849, −15.58441686007880, −15.30066496579330, −14.67269479190140, −13.90571771516960, −13.52046152198437, −13.20268542631522, −12.21091417282409, −11.92046815657448, −11.15657399302716, −10.58243633115838, −10.20856029961735, −9.187956023286037, −8.841250528357901, −8.256695665359497, −7.661743441203943, −7.175489430489124, −6.400271393001587, −5.481212589068981, −5.095623927189608, −4.338217803912152, −3.291507886346988, −2.965791649450365, −2.290019958415553, −1.071940230259338, 0,
1.071940230259338, 2.290019958415553, 2.965791649450365, 3.291507886346988, 4.338217803912152, 5.095623927189608, 5.481212589068981, 6.400271393001587, 7.175489430489124, 7.661743441203943, 8.256695665359497, 8.841250528357901, 9.187956023286037, 10.20856029961735, 10.58243633115838, 11.15657399302716, 11.92046815657448, 12.21091417282409, 13.20268542631522, 13.52046152198437, 13.90571771516960, 14.67269479190140, 15.30066496579330, 15.58441686007880, 16.13752399665849
