| L(s) = 1 | + 2-s + 4-s − 4·5-s + 7-s + 8-s − 3·9-s − 4·10-s − 2·11-s − 6·13-s + 14-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 4·20-s − 2·22-s + 8·23-s + 11·25-s − 6·26-s + 28-s − 8·29-s + 32-s − 6·34-s − 4·35-s − 3·36-s − 2·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.377·7-s + 0.353·8-s − 9-s − 1.26·10-s − 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.894·20-s − 0.426·22-s + 1.66·23-s + 11/5·25-s − 1.17·26-s + 0.188·28-s − 1.48·29-s + 0.176·32-s − 1.02·34-s − 0.676·35-s − 1/2·36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−10.80590269272267363012896143111, −9.282688333154126717687994384109, −8.296202847122805496287857957469, −7.48745095759034509897626872941, −6.90664039006182012073974627909, −5.21625736136705280997603855514, −4.75268745652203605644484325472, −3.51577610402845047123440881457, −2.58082991363778595883241371029, 0,
2.58082991363778595883241371029, 3.51577610402845047123440881457, 4.75268745652203605644484325472, 5.21625736136705280997603855514, 6.90664039006182012073974627909, 7.48745095759034509897626872941, 8.296202847122805496287857957469, 9.282688333154126717687994384109, 10.80590269272267363012896143111
