| L(s) = 1 | + 9·3-s + 78·5-s − 49·7-s + 81·9-s − 444·11-s − 442·13-s + 702·15-s − 126·17-s − 2.68e3·19-s − 441·21-s − 4.20e3·23-s + 2.95e3·25-s + 729·27-s − 5.44e3·29-s − 80·31-s − 3.99e3·33-s − 3.82e3·35-s − 5.43e3·37-s − 3.97e3·39-s + 7.96e3·41-s + 1.15e4·43-s + 6.31e3·45-s + 1.39e4·47-s + 2.40e3·49-s − 1.13e3·51-s − 9.59e3·53-s − 3.46e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.39·5-s − 0.377·7-s + 1/3·9-s − 1.10·11-s − 0.725·13-s + 0.805·15-s − 0.105·17-s − 1.70·19-s − 0.218·21-s − 1.65·23-s + 0.946·25-s + 0.192·27-s − 1.20·29-s − 0.0149·31-s − 0.638·33-s − 0.527·35-s − 0.652·37-s − 0.418·39-s + 0.739·41-s + 0.950·43-s + 0.465·45-s + 0.919·47-s + 1/7·49-s − 0.0610·51-s − 0.469·53-s − 1.54·55-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 78 T + p^{5} T^{2} \) |
| 11 | \( 1 + 444 T + p^{5} T^{2} \) |
| 13 | \( 1 + 34 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 126 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2684 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5442 T + p^{5} T^{2} \) |
| 31 | \( 1 + 80 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5434 T + p^{5} T^{2} \) |
| 41 | \( 1 - 7962 T + p^{5} T^{2} \) |
| 43 | \( 1 - 268 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 13920 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9594 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27492 T + p^{5} T^{2} \) |
| 61 | \( 1 - 49478 T + p^{5} T^{2} \) |
| 67 | \( 1 - 59356 T + p^{5} T^{2} \) |
| 71 | \( 1 + 32040 T + p^{5} T^{2} \) |
| 73 | \( 1 + 61846 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65776 T + p^{5} T^{2} \) |
| 83 | \( 1 + 40188 T + p^{5} T^{2} \) |
| 89 | \( 1 + 7974 T + p^{5} T^{2} \) |
| 97 | \( 1 + 143662 T + p^{5} T^{2} \) |
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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
−10.10430115718847556130352365208, −9.475976354722215030326769534513, −8.471816156807295115973758140692, −7.43480032203906778389900176978, −6.25192840392640455084044631646, −5.44475809352900994440653809888, −4.11105977679246028849055617928, −2.52697664561500972094960474988, −1.98735456118617454596188627028, 0,
1.98735456118617454596188627028, 2.52697664561500972094960474988, 4.11105977679246028849055617928, 5.44475809352900994440653809888, 6.25192840392640455084044631646, 7.43480032203906778389900176978, 8.471816156807295115973758140692, 9.475976354722215030326769534513, 10.10430115718847556130352365208
