| L(s) = 1 | + 3-s + 3·7-s − 2·9-s − 4·11-s − 3·13-s + 3·17-s + 3·21-s − 23-s − 5·25-s − 5·27-s + 29-s − 4·31-s − 4·33-s + 6·37-s − 3·39-s + 4·41-s + 4·47-s + 2·49-s + 3·51-s + 53-s + 9·59-s + 8·61-s − 6·63-s − 3·67-s − 69-s + 12·71-s − 3·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s − 1.20·11-s − 0.832·13-s + 0.727·17-s + 0.654·21-s − 0.208·23-s − 25-s − 0.962·27-s + 0.185·29-s − 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.480·39-s + 0.624·41-s + 0.583·47-s + 2/7·49-s + 0.420·51-s + 0.137·53-s + 1.17·59-s + 1.02·61-s − 0.755·63-s − 0.366·67-s − 0.120·69-s + 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.20523839374834, −13.65616193702963, −13.19918544238006, −12.65141539680077, −12.04852310916337, −11.64073921974084, −11.14943012609414, −10.63811964129656, −10.13909353193140, −9.518135700440761, −9.170937492305097, −8.351287032118673, −8.033281686749651, −7.713133787119328, −7.304004429913904, −6.403252868186838, −5.740048337747601, −5.178003247087542, −5.047925007712124, −4.046054746115549, −3.677057597834902, −2.687839423704084, −2.450909035285375, −1.854559664433806, −0.8892479955006921, 0,
0.8892479955006921, 1.854559664433806, 2.450909035285375, 2.687839423704084, 3.677057597834902, 4.046054746115549, 5.047925007712124, 5.178003247087542, 5.740048337747601, 6.403252868186838, 7.304004429913904, 7.713133787119328, 8.033281686749651, 8.351287032118673, 9.170937492305097, 9.518135700440761, 10.13909353193140, 10.63811964129656, 11.14943012609414, 11.64073921974084, 12.04852310916337, 12.65141539680077, 13.19918544238006, 13.65616193702963, 14.20523839374834
