| L(s) = 1 | − 2-s + 4-s − 3·5-s + 7-s − 8-s + 3·10-s − 3·11-s + 13-s − 14-s + 16-s + 3·17-s + 5·19-s − 3·20-s + 3·22-s + 3·23-s + 4·25-s − 26-s + 28-s + 3·29-s − 10·31-s − 32-s − 3·34-s − 3·35-s − 7·37-s − 5·38-s + 3·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.904·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s − 0.670·20-s + 0.639·22-s + 0.625·23-s + 4/5·25-s − 0.196·26-s + 0.188·28-s + 0.557·29-s − 1.79·31-s − 0.176·32-s − 0.514·34-s − 0.507·35-s − 1.15·37-s − 0.811·38-s + 0.474·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.851633452241949887706170327808, −8.091285142177144740601668535575, −7.56353523982414710439528855608, −7.01142907917938592333952322993, −5.65940843144591511168796809859, −4.89898920843487998926105208871, −3.68091448723354985781129327882, −2.95150623961030889311106388300, −1.40498596438442826877223932759, 0,
1.40498596438442826877223932759, 2.95150623961030889311106388300, 3.68091448723354985781129327882, 4.89898920843487998926105208871, 5.65940843144591511168796809859, 7.01142907917938592333952322993, 7.56353523982414710439528855608, 8.091285142177144740601668535575, 8.851633452241949887706170327808
