| L(s) = 1 | − 2·5-s − 2·11-s + 3·13-s + 8·17-s + 19-s − 8·23-s − 25-s − 4·29-s − 3·31-s − 37-s + 6·41-s + 11·43-s + 6·47-s + 12·53-s + 4·55-s + 4·59-s + 6·61-s − 6·65-s + 13·67-s + 10·71-s + 11·73-s − 3·79-s + 2·83-s − 16·85-s − 2·95-s − 10·97-s + 10·101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.603·11-s + 0.832·13-s + 1.94·17-s + 0.229·19-s − 1.66·23-s − 1/5·25-s − 0.742·29-s − 0.538·31-s − 0.164·37-s + 0.937·41-s + 1.67·43-s + 0.875·47-s + 1.64·53-s + 0.539·55-s + 0.520·59-s + 0.768·61-s − 0.744·65-s + 1.58·67-s + 1.18·71-s + 1.28·73-s − 0.337·79-s + 0.219·83-s − 1.73·85-s − 0.205·95-s − 1.01·97-s + 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.391509355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.391509355\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.340766954255413624828972920362, −8.213177915144848467457073975549, −7.86003472307450153648690839574, −7.14892757864943782004910226592, −5.84853197538479925165694215764, −5.44492897197805836950949219523, −3.97239648734814585129569787000, −3.65955604277278655695987739777, −2.31898745928895941643396981279, −0.814330480313867343133277258395,
0.814330480313867343133277258395, 2.31898745928895941643396981279, 3.65955604277278655695987739777, 3.97239648734814585129569787000, 5.44492897197805836950949219523, 5.84853197538479925165694215764, 7.14892757864943782004910226592, 7.86003472307450153648690839574, 8.213177915144848467457073975549, 9.340766954255413624828972920362
