| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 4·11-s + 16-s − 2·17-s − 6·19-s − 20-s + 4·22-s + 2·23-s + 25-s + 6·29-s + 32-s − 2·34-s + 6·37-s − 6·38-s − 40-s + 2·41-s + 4·44-s + 2·46-s − 2·47-s − 7·49-s + 50-s + 4·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 1.20·11-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.223·20-s + 0.852·22-s + 0.417·23-s + 1/5·25-s + 1.11·29-s + 0.176·32-s − 0.342·34-s + 0.986·37-s − 0.973·38-s − 0.158·40-s + 0.312·41-s + 0.603·44-s + 0.294·46-s − 0.291·47-s − 49-s + 0.141·50-s + 0.549·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.32268158402221, −13.02154680481202, −12.58972684773179, −12.08402237285468, −11.56405445528585, −11.27440019084631, −10.84229486040456, −10.11468083686086, −9.850958464629434, −8.963877362567825, −8.729085491847540, −8.218837586968273, −7.584950963720820, −6.979822885151343, −6.609132624175076, −6.194845149084754, −5.686631653608590, −4.791773653372408, −4.563006992436820, −3.980579581849128, −3.604440949464853, −2.788094871199842, −2.377104902506140, −1.547416453816399, −0.9670072862468594, 0,
0.9670072862468594, 1.547416453816399, 2.377104902506140, 2.788094871199842, 3.604440949464853, 3.980579581849128, 4.563006992436820, 4.791773653372408, 5.686631653608590, 6.194845149084754, 6.609132624175076, 6.979822885151343, 7.584950963720820, 8.218837586968273, 8.729085491847540, 8.963877362567825, 9.850958464629434, 10.11468083686086, 10.84229486040456, 11.27440019084631, 11.56405445528585, 12.08402237285468, 12.58972684773179, 13.02154680481202, 13.32268158402221
