| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 3·11-s + 13-s − 15-s + 7·17-s − 19-s − 2·21-s − 3·23-s + 25-s + 27-s + 3·29-s + 9·31-s + 3·33-s + 2·35-s − 37-s + 39-s − 6·41-s − 4·43-s − 45-s + 2·47-s − 3·49-s + 7·51-s − 6·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s + 1.69·17-s − 0.229·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 1.61·31-s + 0.522·33-s + 0.338·35-s − 0.164·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.980·51-s − 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−15.06345669670102, −14.45741226694928, −13.99688985587050, −13.65186037546110, −12.96032534001195, −12.33155446051327, −11.99150584866596, −11.64469612944169, −10.64824667519829, −10.29062036775199, −9.651912918180220, −9.352708634502936, −8.536181448592164, −8.173111014550596, −7.623492500616832, −6.970887357409023, −6.312894254290252, −6.033860423844226, −5.051693432826223, −4.403971919181304, −3.807273836260126, −3.141007269708962, −2.898190616139972, −1.656909141047227, −1.130140947396113, 0,
1.130140947396113, 1.656909141047227, 2.898190616139972, 3.141007269708962, 3.807273836260126, 4.403971919181304, 5.051693432826223, 6.033860423844226, 6.312894254290252, 6.970887357409023, 7.623492500616832, 8.173111014550596, 8.536181448592164, 9.352708634502936, 9.651912918180220, 10.29062036775199, 10.64824667519829, 11.64469612944169, 11.99150584866596, 12.33155446051327, 12.96032534001195, 13.65186037546110, 13.99688985587050, 14.45741226694928, 15.06345669670102
