| L(s) = 1 | − 9·3-s − 72·5-s − 49·7-s + 81·9-s + 414·11-s − 1.05e3·13-s + 648·15-s − 1.84e3·17-s − 236·19-s + 441·21-s − 2.89e3·23-s + 2.05e3·25-s − 729·27-s − 6.52e3·29-s − 6.20e3·31-s − 3.72e3·33-s + 3.52e3·35-s + 9.65e3·37-s + 9.48e3·39-s + 8.48e3·41-s + 1.08e4·43-s − 5.83e3·45-s − 60·47-s + 2.40e3·49-s + 1.66e4·51-s + 2.25e4·53-s − 2.98e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.28·5-s − 0.377·7-s + 1/3·9-s + 1.03·11-s − 1.72·13-s + 0.743·15-s − 1.55·17-s − 0.149·19-s + 0.218·21-s − 1.14·23-s + 0.658·25-s − 0.192·27-s − 1.44·29-s − 1.15·31-s − 0.595·33-s + 0.486·35-s + 1.15·37-s + 0.998·39-s + 0.788·41-s + 0.891·43-s − 0.429·45-s − 0.00396·47-s + 1/7·49-s + 0.895·51-s + 1.10·53-s − 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4206806668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4206806668\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 + 72 T + p^{5} T^{2} \) |
| 11 | \( 1 - 414 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1054 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1848 T + p^{5} T^{2} \) |
| 19 | \( 1 + 236 T + p^{5} T^{2} \) |
| 23 | \( 1 + 126 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 6522 T + p^{5} T^{2} \) |
| 31 | \( 1 + 200 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 9650 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8484 T + p^{5} T^{2} \) |
| 43 | \( 1 - 10804 T + p^{5} T^{2} \) |
| 47 | \( 1 + 60 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22506 T + p^{5} T^{2} \) |
| 59 | \( 1 - 28176 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35194 T + p^{5} T^{2} \) |
| 67 | \( 1 - 28216 T + p^{5} T^{2} \) |
| 71 | \( 1 - 6642 T + p^{5} T^{2} \) |
| 73 | \( 1 + 52090 T + p^{5} T^{2} \) |
| 79 | \( 1 + 43340 T + p^{5} T^{2} \) |
| 83 | \( 1 + 25716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 98724 T + p^{5} T^{2} \) |
| 97 | \( 1 + 148954 T + p^{5} T^{2} \) |
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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
−11.00430131934165553943639578198, −9.762920202848843612610408946788, −8.950016091760625173647873798910, −7.60390758863762244147552076672, −7.04575413722676619429245451097, −5.89323204312664808091256824340, −4.48024352286451848087508673861, −3.88492651217111282729955227698, −2.21388986664691716492940777354, −0.34786322798922030555320398999,
0.34786322798922030555320398999, 2.21388986664691716492940777354, 3.88492651217111282729955227698, 4.48024352286451848087508673861, 5.89323204312664808091256824340, 7.04575413722676619429245451097, 7.60390758863762244147552076672, 8.950016091760625173647873798910, 9.762920202848843612610408946788, 11.00430131934165553943639578198
