| L(s) = 1 | − 3-s − 2·5-s − 2·9-s + 11-s + 5·13-s + 2·15-s + 17-s + 6·19-s − 25-s + 5·27-s + 6·29-s − 4·31-s − 33-s − 8·37-s − 5·39-s − 6·41-s + 12·43-s + 4·45-s + 2·47-s − 51-s + 7·53-s − 2·55-s − 6·57-s − 12·59-s − 12·61-s − 10·65-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 2/3·9-s + 0.301·11-s + 1.38·13-s + 0.516·15-s + 0.242·17-s + 1.37·19-s − 1/5·25-s + 0.962·27-s + 1.11·29-s − 0.718·31-s − 0.174·33-s − 1.31·37-s − 0.800·39-s − 0.937·41-s + 1.82·43-s + 0.596·45-s + 0.291·47-s − 0.140·51-s + 0.961·53-s − 0.269·55-s − 0.794·57-s − 1.56·59-s − 1.53·61-s − 1.24·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−14.66054070701909, −14.00908605074533, −13.87337804372578, −13.23686112201198, −12.33572767796141, −12.09728918124691, −11.70653399394627, −11.16366284482325, −10.69184004745281, −10.33030873532179, −9.369514466685821, −8.967156132485855, −8.475137867714591, −7.799523420516048, −7.477955381630150, −6.637694056797004, −6.238909661596852, −5.559461303294343, −5.171489187829456, −4.384096306691299, −3.684231199076587, −3.351530065542080, −2.607924161301422, −1.473028251796908, −0.8906174845541227, 0,
0.8906174845541227, 1.473028251796908, 2.607924161301422, 3.351530065542080, 3.684231199076587, 4.384096306691299, 5.171489187829456, 5.559461303294343, 6.238909661596852, 6.637694056797004, 7.477955381630150, 7.799523420516048, 8.475137867714591, 8.967156132485855, 9.369514466685821, 10.33030873532179, 10.69184004745281, 11.16366284482325, 11.70653399394627, 12.09728918124691, 12.33572767796141, 13.23686112201198, 13.87337804372578, 14.00908605074533, 14.66054070701909
