| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s + 6·13-s + 15-s + 6·17-s + 4·19-s − 8·23-s + 25-s + 27-s − 10·29-s − 4·31-s − 4·33-s + 6·37-s + 6·39-s − 6·41-s − 4·43-s + 45-s + 12·47-s + 6·51-s − 6·53-s − 4·55-s + 4·57-s + 4·59-s − 2·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.696·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 1.75·47-s + 0.840·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.82277530906663, −14.21215370660140, −13.76240472684741, −13.46001712333239, −12.87982680925373, −12.45351463496031, −11.71809921994634, −11.19996193564825, −10.60776026861958, −10.12431710539949, −9.626742396100439, −9.190725735058986, −8.361792613821901, −8.077023341526268, −7.540856429001850, −7.015299995743845, −6.049528102176763, −5.590863484708768, −5.432293427730822, −4.287249693420278, −3.725240499206707, −3.227336453965431, −2.545108477002756, −1.707680509236774, −1.218707297086455, 0,
1.218707297086455, 1.707680509236774, 2.545108477002756, 3.227336453965431, 3.725240499206707, 4.287249693420278, 5.432293427730822, 5.590863484708768, 6.049528102176763, 7.015299995743845, 7.540856429001850, 8.077023341526268, 8.361792613821901, 9.190725735058986, 9.626742396100439, 10.12431710539949, 10.60776026861958, 11.19996193564825, 11.71809921994634, 12.45351463496031, 12.87982680925373, 13.46001712333239, 13.76240472684741, 14.21215370660140, 14.82277530906663
