| L(s) = 1 | − 38·5-s + 120·7-s − 524·11-s − 962·13-s + 1.35e3·17-s − 2.28e3·19-s − 2.55e3·23-s − 1.68e3·25-s − 3.96e3·29-s − 2.99e3·31-s − 4.56e3·35-s + 1.32e4·37-s + 1.51e4·41-s − 7.31e3·43-s + 6.96e3·47-s − 2.40e3·49-s + 1.74e4·53-s + 1.99e4·55-s − 3.38e4·59-s + 3.91e4·61-s + 3.65e4·65-s + 3.29e4·67-s − 1.42e4·71-s − 3.59e4·73-s − 6.28e4·77-s − 2.98e4·79-s + 5.18e4·83-s + ⋯ |
| L(s) = 1 | − 0.679·5-s + 0.925·7-s − 1.30·11-s − 1.57·13-s + 1.13·17-s − 1.45·19-s − 1.00·23-s − 0.537·25-s − 0.875·29-s − 0.559·31-s − 0.629·35-s + 1.58·37-s + 1.40·41-s − 0.603·43-s + 0.459·47-s − 0.143·49-s + 0.854·53-s + 0.887·55-s − 1.26·59-s + 1.34·61-s + 1.07·65-s + 0.897·67-s − 0.335·71-s − 0.790·73-s − 1.20·77-s − 0.538·79-s + 0.826·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 38 T + p^{5} T^{2} \) |
| 7 | \( 1 - 120 T + p^{5} T^{2} \) |
| 11 | \( 1 + 524 T + p^{5} T^{2} \) |
| 13 | \( 1 + 74 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 1358 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2284 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2552 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3966 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2992 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13206 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15126 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7316 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6960 T + p^{5} T^{2} \) |
| 53 | \( 1 - 17482 T + p^{5} T^{2} \) |
| 59 | \( 1 + 33884 T + p^{5} T^{2} \) |
| 61 | \( 1 - 39118 T + p^{5} T^{2} \) |
| 67 | \( 1 - 32996 T + p^{5} T^{2} \) |
| 71 | \( 1 + 14248 T + p^{5} T^{2} \) |
| 73 | \( 1 + 35990 T + p^{5} T^{2} \) |
| 79 | \( 1 + 29888 T + p^{5} T^{2} \) |
| 83 | \( 1 - 51884 T + p^{5} T^{2} \) |
| 89 | \( 1 + 30714 T + p^{5} T^{2} \) |
| 97 | \( 1 + 48478 T + p^{5} T^{2} \) |
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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
−12.94188582402373277997310027284, −12.01578695287933972554689697506, −10.91493290038787885623878042276, −9.807408883798789183684865736954, −8.062783807620532864629447281755, −7.56205594808745117508091576140, −5.53267693503199113880753094523, −4.28549938294476363477904834702, −2.32416805286548267303816400384, 0,
2.32416805286548267303816400384, 4.28549938294476363477904834702, 5.53267693503199113880753094523, 7.56205594808745117508091576140, 8.062783807620532864629447281755, 9.807408883798789183684865736954, 10.91493290038787885623878042276, 12.01578695287933972554689697506, 12.94188582402373277997310027284
