| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2·13-s + 15-s + 2·17-s + 21-s − 23-s + 25-s + 27-s − 10·29-s + 4·31-s + 35-s + 6·37-s + 2·39-s + 10·41-s + 12·43-s + 45-s + 49-s + 2·51-s − 6·53-s − 4·59-s + 14·61-s + 63-s + 2·65-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.218·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 0.320·39-s + 1.56·41-s + 1.82·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.520·59-s + 1.79·61-s + 0.125·63-s + 0.248·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.135129781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.135129781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−14.62030040527762, −14.34546993499306, −13.89616832175642, −13.31310031541891, −12.66725659276410, −12.54195709318941, −11.55779200997102, −11.03732372815442, −10.79490240122463, −9.829253363519792, −9.514418278956836, −9.125496561556231, −8.223340335146920, −8.017356110147289, −7.361482830424264, −6.728612755351271, −5.989690865513487, −5.581893556606060, −4.895006247639748, −3.962864707542766, −3.816795563621276, −2.702758439695906, −2.324629655111964, −1.446765598480879, −0.7611146807439646,
0.7611146807439646, 1.446765598480879, 2.324629655111964, 2.702758439695906, 3.816795563621276, 3.962864707542766, 4.895006247639748, 5.581893556606060, 5.989690865513487, 6.728612755351271, 7.361482830424264, 8.017356110147289, 8.223340335146920, 9.125496561556231, 9.514418278956836, 9.829253363519792, 10.79490240122463, 11.03732372815442, 11.55779200997102, 12.54195709318941, 12.66725659276410, 13.31310031541891, 13.89616832175642, 14.34546993499306, 14.62030040527762
