| L(s) = 1 | + 9·3-s − 66·5-s − 176·7-s + 81·9-s + 60·11-s − 658·13-s − 594·15-s − 414·17-s − 956·19-s − 1.58e3·21-s − 600·23-s + 1.23e3·25-s + 729·27-s + 5.57e3·29-s + 3.59e3·31-s + 540·33-s + 1.16e4·35-s − 8.45e3·37-s − 5.92e3·39-s + 1.91e4·41-s − 1.33e4·43-s − 5.34e3·45-s + 1.96e4·47-s + 1.41e4·49-s − 3.72e3·51-s − 3.12e4·53-s − 3.96e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.18·5-s − 1.35·7-s + 1/3·9-s + 0.149·11-s − 1.07·13-s − 0.681·15-s − 0.347·17-s − 0.607·19-s − 0.783·21-s − 0.236·23-s + 0.393·25-s + 0.192·27-s + 1.23·29-s + 0.671·31-s + 0.0863·33-s + 1.60·35-s − 1.01·37-s − 0.623·39-s + 1.78·41-s − 1.09·43-s − 0.393·45-s + 1.29·47-s + 0.843·49-s − 0.200·51-s − 1.52·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 66 T + p^{5} T^{2} \) |
| 7 | \( 1 + 176 T + p^{5} T^{2} \) |
| 11 | \( 1 - 60 T + p^{5} T^{2} \) |
| 13 | \( 1 + 658 T + p^{5} T^{2} \) |
| 17 | \( 1 + 414 T + p^{5} T^{2} \) |
| 19 | \( 1 + 956 T + p^{5} T^{2} \) |
| 23 | \( 1 + 600 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5574 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3592 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8458 T + p^{5} T^{2} \) |
| 41 | \( 1 - 19194 T + p^{5} T^{2} \) |
| 43 | \( 1 + 13316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 19680 T + p^{5} T^{2} \) |
| 53 | \( 1 + 31266 T + p^{5} T^{2} \) |
| 59 | \( 1 + 26340 T + p^{5} T^{2} \) |
| 61 | \( 1 + 31090 T + p^{5} T^{2} \) |
| 67 | \( 1 - 16804 T + p^{5} T^{2} \) |
| 71 | \( 1 + 6120 T + p^{5} T^{2} \) |
| 73 | \( 1 + 25558 T + p^{5} T^{2} \) |
| 79 | \( 1 + 74408 T + p^{5} T^{2} \) |
| 83 | \( 1 - 6468 T + p^{5} T^{2} \) |
| 89 | \( 1 + 32742 T + p^{5} T^{2} \) |
| 97 | \( 1 - 166082 T + p^{5} T^{2} \) |
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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
−14.15264371278602592019456279786, −12.79212769532995176276559275717, −11.98215101287879823547544206284, −10.34493145693428254395256167319, −9.141743676360213261126896381840, −7.80583564714552416227920490863, −6.60544577993134557346225331985, −4.30266278584201242570074958133, −2.92437735962115777125174653471, 0,
2.92437735962115777125174653471, 4.30266278584201242570074958133, 6.60544577993134557346225331985, 7.80583564714552416227920490863, 9.141743676360213261126896381840, 10.34493145693428254395256167319, 11.98215101287879823547544206284, 12.79212769532995176276559275717, 14.15264371278602592019456279786
