| L(s) = 1 | + 2·2-s − 4·4-s − 5-s − 7·7-s − 24·8-s − 2·10-s + 11·11-s + 7·13-s − 14·14-s − 16·16-s + 14·17-s − 45·19-s + 4·20-s + 22·22-s + 88·23-s − 124·25-s + 14·26-s + 28·28-s + 69·29-s + 22·31-s + 160·32-s + 28·34-s + 7·35-s + 57·37-s − 90·38-s + 24·40-s + 380·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.0894·5-s − 0.377·7-s − 1.06·8-s − 0.0632·10-s + 0.301·11-s + 0.149·13-s − 0.267·14-s − 1/4·16-s + 0.199·17-s − 0.543·19-s + 0.0447·20-s + 0.213·22-s + 0.797·23-s − 0.991·25-s + 0.105·26-s + 0.188·28-s + 0.441·29-s + 0.127·31-s + 0.883·32-s + 0.141·34-s + 0.0338·35-s + 0.253·37-s − 0.384·38-s + 0.0948·40-s + 1.44·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.991144552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.991144552\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 5 | \( 1 + T + p^{3} T^{2} \) |
| 13 | \( 1 - 7 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 45 T + p^{3} T^{2} \) |
| 23 | \( 1 - 88 T + p^{3} T^{2} \) |
| 29 | \( 1 - 69 T + p^{3} T^{2} \) |
| 31 | \( 1 - 22 T + p^{3} T^{2} \) |
| 37 | \( 1 - 57 T + p^{3} T^{2} \) |
| 41 | \( 1 - 380 T + p^{3} T^{2} \) |
| 43 | \( 1 - 48 T + p^{3} T^{2} \) |
| 47 | \( 1 - 385 T + p^{3} T^{2} \) |
| 53 | \( 1 - 672 T + p^{3} T^{2} \) |
| 59 | \( 1 - 469 T + p^{3} T^{2} \) |
| 61 | \( 1 + 342 T + p^{3} T^{2} \) |
| 67 | \( 1 + 139 T + p^{3} T^{2} \) |
| 71 | \( 1 + 132 T + p^{3} T^{2} \) |
| 73 | \( 1 - 145 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1244 T + p^{3} T^{2} \) |
| 83 | \( 1 + 522 T + p^{3} T^{2} \) |
| 89 | \( 1 + 822 T + p^{3} T^{2} \) |
| 97 | \( 1 - 272 T + p^{3} T^{2} \) |
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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
−9.987994068731365345071473404866, −9.186896069311863973039843929473, −8.473894102148616565102028162203, −7.34223425182523230787220981400, −6.25905327476354122980015967423, −5.53730142149248904993793084082, −4.41858659939594583211532429344, −3.70021117167718319732584525304, −2.55313846555331212411806747010, −0.73257860969446132913644639821,
0.73257860969446132913644639821, 2.55313846555331212411806747010, 3.70021117167718319732584525304, 4.41858659939594583211532429344, 5.53730142149248904993793084082, 6.25905327476354122980015967423, 7.34223425182523230787220981400, 8.473894102148616565102028162203, 9.186896069311863973039843929473, 9.987994068731365345071473404866
