| L(s) = 1 | − 16·5-s + 49·7-s + 8·11-s + 684·13-s + 2.21e3·17-s + 2.69e3·19-s + 3.34e3·23-s − 2.86e3·25-s + 3.25e3·29-s − 4.78e3·31-s − 784·35-s − 1.14e4·37-s − 1.33e4·41-s + 928·43-s + 1.21e3·47-s + 2.40e3·49-s − 1.31e4·53-s − 128·55-s + 3.47e4·59-s − 1.03e3·61-s − 1.09e4·65-s − 1.01e4·67-s + 6.27e4·71-s − 1.89e4·73-s + 392·77-s − 1.14e4·79-s + 8.89e4·83-s + ⋯ |
| L(s) = 1 | − 0.286·5-s + 0.377·7-s + 0.0199·11-s + 1.12·13-s + 1.86·17-s + 1.71·19-s + 1.31·23-s − 0.918·25-s + 0.718·29-s − 0.894·31-s − 0.108·35-s − 1.37·37-s − 1.24·41-s + 0.0765·43-s + 0.0800·47-s + 1/7·49-s − 0.641·53-s − 0.00570·55-s + 1.29·59-s − 0.0355·61-s − 0.321·65-s − 0.275·67-s + 1.47·71-s − 0.415·73-s + 0.00753·77-s − 0.205·79-s + 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.895836091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.895836091\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 16 T + p^{5} T^{2} \) |
| 11 | \( 1 - 8 T + p^{5} T^{2} \) |
| 13 | \( 1 - 684 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2218 T + p^{5} T^{2} \) |
| 19 | \( 1 - 142 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 3344 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3254 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4788 T + p^{5} T^{2} \) |
| 37 | \( 1 + 310 p T + p^{5} T^{2} \) |
| 41 | \( 1 + 13350 T + p^{5} T^{2} \) |
| 43 | \( 1 - 928 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1212 T + p^{5} T^{2} \) |
| 53 | \( 1 + 13110 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34702 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1032 T + p^{5} T^{2} \) |
| 67 | \( 1 + 10108 T + p^{5} T^{2} \) |
| 71 | \( 1 - 62720 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18926 T + p^{5} T^{2} \) |
| 79 | \( 1 + 11400 T + p^{5} T^{2} \) |
| 83 | \( 1 - 88958 T + p^{5} T^{2} \) |
| 89 | \( 1 + 19722 T + p^{5} T^{2} \) |
| 97 | \( 1 - 17062 T + p^{5} T^{2} \) |
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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
−9.226934026213564220404654216513, −8.310171788116802114004245628619, −7.63624201832512366070432570325, −6.82147028698586097847382953816, −5.59506657500165321599204288062, −5.11214129918270669945083502015, −3.67553000911988005955994204467, −3.18558562721220701115837533185, −1.56574096074611512529859980362, −0.815253984644625904741777780370,
0.815253984644625904741777780370, 1.56574096074611512529859980362, 3.18558562721220701115837533185, 3.67553000911988005955994204467, 5.11214129918270669945083502015, 5.59506657500165321599204288062, 6.82147028698586097847382953816, 7.63624201832512366070432570325, 8.310171788116802114004245628619, 9.226934026213564220404654216513
