| L(s) = 1 | + 2-s + 2·3-s − 4-s − 4·5-s + 2·6-s + 7-s − 3·8-s + 9-s − 4·10-s − 4·11-s − 2·12-s − 4·13-s + 14-s − 8·15-s − 16-s − 6·17-s + 18-s + 6·19-s + 4·20-s + 2·21-s − 4·22-s − 8·23-s − 6·24-s + 11·25-s − 4·26-s − 4·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 1.26·10-s − 1.20·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 2.06·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.37·19-s + 0.894·20-s + 0.436·21-s − 0.852·22-s − 1.66·23-s − 1.22·24-s + 11/5·25-s − 0.784·26-s − 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.211459780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211459780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.947375272568936419893589805758, −7.78519757983091297538361364711, −6.86817419715771345470615286955, −5.70604871297901348895477432051, −4.78917356910861509236595307513, −4.40160707744262449084047360422, −3.73386956696739777342200091619, −2.86099928219130698985650692088, −2.46340290650082717059423174408, −0.46004150442772757080641625620,
0.46004150442772757080641625620, 2.46340290650082717059423174408, 2.86099928219130698985650692088, 3.73386956696739777342200091619, 4.40160707744262449084047360422, 4.78917356910861509236595307513, 5.70604871297901348895477432051, 6.86817419715771345470615286955, 7.78519757983091297538361364711, 7.947375272568936419893589805758
