| L(s) = 1 | + 3-s − 4·5-s − 7-s + 9-s − 2·11-s + 2·13-s − 4·15-s + 2·17-s + 2·19-s − 21-s − 23-s + 11·25-s + 27-s − 6·29-s − 2·33-s + 4·35-s + 4·37-s + 2·39-s − 10·41-s + 10·43-s − 4·45-s + 8·47-s + 49-s + 2·51-s + 8·55-s + 2·57-s − 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 0.348·33-s + 0.676·35-s + 0.657·37-s + 0.320·39-s − 1.56·41-s + 1.52·43-s − 0.596·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 1.07·55-s + 0.264·57-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.56914376814153051010772356046, −7.19505884100231513035605738206, −6.20232109450845556998997367296, −5.33104556274669681253666068719, −4.45403955364002645103432993227, −3.76176959720027725991825598035, −3.32577856313445476099193287697, −2.48204174595099434538927314311, −1.11127285090253336780912946476, 0,
1.11127285090253336780912946476, 2.48204174595099434538927314311, 3.32577856313445476099193287697, 3.76176959720027725991825598035, 4.45403955364002645103432993227, 5.33104556274669681253666068719, 6.20232109450845556998997367296, 7.19505884100231513035605738206, 7.56914376814153051010772356046
