L(s) = 1 | − 3-s + 2·5-s + 9-s − 2·13-s − 2·15-s − 2·17-s + 4·19-s + 8·23-s − 25-s − 27-s + 6·29-s − 8·31-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s + 2·53-s − 4·57-s + 4·59-s − 2·61-s − 4·65-s − 4·67-s − 8·69-s − 8·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s − 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−15.82990475860147, −15.17216014977464, −14.64276871927380, −14.09392465953872, −13.51533669850880, −13.02760940167935, −12.57476071682863, −11.87860951167232, −11.39509703585013, −10.78638377778716, −10.27740290449854, −9.703161057728960, −9.199021555216559, −8.696453798989065, −7.820487550732530, −7.046191741606810, −6.863092616431083, −5.941952593884288, −5.490245924050042, −4.944548997435659, −4.339902203589440, −3.320450658683481, −2.711366608873740, −1.809570618965464, −1.142034592903933, 0,
1.142034592903933, 1.809570618965464, 2.711366608873740, 3.320450658683481, 4.339902203589440, 4.944548997435659, 5.490245924050042, 5.941952593884288, 6.863092616431083, 7.046191741606810, 7.820487550732530, 8.696453798989065, 9.199021555216559, 9.703161057728960, 10.27740290449854, 10.78638377778716, 11.39509703585013, 11.87860951167232, 12.57476071682863, 13.02760940167935, 13.51533669850880, 14.09392465953872, 14.64276871927380, 15.17216014977464, 15.82990475860147