L(s) = 1 | + 3-s + 2·5-s − 2·7-s + 9-s + 11-s − 2·13-s + 2·15-s + 4·17-s − 6·19-s − 2·21-s − 25-s + 27-s − 8·29-s − 8·31-s + 33-s − 4·35-s + 10·37-s − 2·39-s + 8·41-s − 2·43-s + 2·45-s − 8·47-s − 3·49-s + 4·51-s − 2·53-s + 2·55-s − 6·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 0.174·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 1.24·41-s − 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.269·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.328164578\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328164578\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.10795468637363568116956190325, −12.70566797794869228083627204121, −11.13297096888404845423920902820, −9.796917351475791676316160488949, −9.413820553443583654003786378045, −7.996057650348538062805293058353, −6.70453643193197308588262041684, −5.57273521417705379772496968279, −3.79294235641559942977546839028, −2.21252199435586523776406167130,
2.21252199435586523776406167130, 3.79294235641559942977546839028, 5.57273521417705379772496968279, 6.70453643193197308588262041684, 7.996057650348538062805293058353, 9.413820553443583654003786378045, 9.796917351475791676316160488949, 11.13297096888404845423920902820, 12.70566797794869228083627204121, 13.10795468637363568116956190325