| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 12-s + 4·13-s − 14-s + 16-s + 18-s + 2·19-s + 21-s − 23-s − 24-s + 4·26-s − 27-s − 28-s + 6·29-s + 2·31-s + 32-s + 36-s + 10·37-s + 2·38-s − 4·39-s + 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 0.218·21-s − 0.208·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 1/6·36-s + 1.64·37-s + 0.324·38-s − 0.640·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.378207239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.378207239\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−15.58553313953578, −14.78516038857855, −14.37374294225517, −13.58463657391276, −13.36237358462708, −12.71711756997080, −12.24173871768397, −11.55608865324984, −11.29214200451084, −10.56849329222487, −10.12102973051599, −9.449851128558886, −8.792356539239740, −8.034256674728996, −7.540880126678250, −6.647427679833880, −6.374136337843288, −5.725988808568159, −5.229471081262948, −4.291970797925612, −4.039339234546002, −3.080869337983016, −2.538039808041471, −1.416324614901313, −0.7298327354466679,
0.7298327354466679, 1.416324614901313, 2.538039808041471, 3.080869337983016, 4.039339234546002, 4.291970797925612, 5.229471081262948, 5.725988808568159, 6.374136337843288, 6.647427679833880, 7.540880126678250, 8.034256674728996, 8.792356539239740, 9.449851128558886, 10.12102973051599, 10.56849329222487, 11.29214200451084, 11.55608865324984, 12.24173871768397, 12.71711756997080, 13.36237358462708, 13.58463657391276, 14.37374294225517, 14.78516038857855, 15.58553313953578
