| L(s) = 1 | + 6·2-s + 4·4-s + 78·5-s − 168·8-s + 468·10-s − 444·11-s + 442·13-s − 1.13e3·16-s − 126·17-s − 2.68e3·19-s + 312·20-s − 2.66e3·22-s − 4.20e3·23-s + 2.95e3·25-s + 2.65e3·26-s + 5.44e3·29-s − 80·31-s − 1.44e3·32-s − 756·34-s − 5.43e3·37-s − 1.61e4·38-s − 1.31e4·40-s + 7.96e3·41-s − 1.15e4·43-s − 1.77e3·44-s − 2.52e4·46-s − 1.39e4·47-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s + 1.39·5-s − 0.928·8-s + 1.47·10-s − 1.10·11-s + 0.725·13-s − 1.10·16-s − 0.105·17-s − 1.70·19-s + 0.174·20-s − 1.17·22-s − 1.65·23-s + 0.946·25-s + 0.769·26-s + 1.20·29-s − 0.0149·31-s − 0.248·32-s − 0.112·34-s − 0.652·37-s − 1.80·38-s − 1.29·40-s + 0.739·41-s − 0.950·43-s − 0.138·44-s − 1.75·46-s − 0.919·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 p T + p^{5} T^{2} \) |
| 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
−10.07692297478818750775621076663, −8.956595533049438268522261144138, −8.161445448642149201093113571389, −6.45726178747999662844777730689, −5.99938426150470216725761353316, −5.09211345667075973013916800207, −4.12337393221556162501178185901, −2.79650910170453002604329227238, −1.87565498961155357665666407582, 0,
1.87565498961155357665666407582, 2.79650910170453002604329227238, 4.12337393221556162501178185901, 5.09211345667075973013916800207, 5.99938426150470216725761353316, 6.45726178747999662844777730689, 8.161445448642149201093113571389, 8.956595533049438268522261144138, 10.07692297478818750775621076663
