| L(s) = 1 | + 7·5-s + 13·7-s − 26·11-s + 13·13-s − 77·17-s + 126·19-s − 96·23-s − 76·25-s + 82·29-s − 196·31-s + 91·35-s − 131·37-s − 336·41-s + 201·43-s − 105·47-s − 174·49-s + 432·53-s − 182·55-s − 294·59-s − 56·61-s + 91·65-s − 478·67-s + 9·71-s + 98·73-s − 338·77-s − 1.30e3·79-s − 308·83-s + ⋯ |
| L(s) = 1 | + 0.626·5-s + 0.701·7-s − 0.712·11-s + 0.277·13-s − 1.09·17-s + 1.52·19-s − 0.870·23-s − 0.607·25-s + 0.525·29-s − 1.13·31-s + 0.439·35-s − 0.582·37-s − 1.27·41-s + 0.712·43-s − 0.325·47-s − 0.507·49-s + 1.11·53-s − 0.446·55-s − 0.648·59-s − 0.117·61-s + 0.173·65-s − 0.871·67-s + 0.0150·71-s + 0.157·73-s − 0.500·77-s − 1.85·79-s − 0.407·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 17 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 - 126 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 131 T + p^{3} T^{2} \) |
| 41 | \( 1 + 336 T + p^{3} T^{2} \) |
| 43 | \( 1 - 201 T + p^{3} T^{2} \) |
| 47 | \( 1 + 105 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 + 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 56 T + p^{3} T^{2} \) |
| 67 | \( 1 + 478 T + p^{3} T^{2} \) |
| 71 | \( 1 - 9 T + p^{3} T^{2} \) |
| 73 | \( 1 - 98 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 + 308 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−8.480086513468905640578713219306, −7.72213909623913587015032488339, −6.96721111218392620606767480054, −5.91727302272227481488266734350, −5.31636048711874553081784215114, −4.47356408088371820942110661105, −3.36481540098648956502320318929, −2.25163826307861076138383214535, −1.45884912883188450472695849084, 0,
1.45884912883188450472695849084, 2.25163826307861076138383214535, 3.36481540098648956502320318929, 4.47356408088371820942110661105, 5.31636048711874553081784215114, 5.91727302272227481488266734350, 6.96721111218392620606767480054, 7.72213909623913587015032488339, 8.480086513468905640578713219306
