L(s) = 1 | − 3-s + 5-s + 9-s − 3·11-s − 4·13-s − 15-s + 4·19-s + 8·23-s − 4·25-s − 27-s + 3·29-s − 5·31-s + 3·33-s − 8·37-s + 4·39-s + 8·41-s − 6·43-s + 45-s + 10·47-s − 9·53-s − 3·55-s − 4·57-s + 5·59-s + 10·61-s − 4·65-s − 6·67-s − 8·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.258·15-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.898·31-s + 0.522·33-s − 1.31·37-s + 0.640·39-s + 1.24·41-s − 0.914·43-s + 0.149·45-s + 1.45·47-s − 1.23·53-s − 0.404·55-s − 0.529·57-s + 0.650·59-s + 1.28·61-s − 0.496·65-s − 0.733·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.25556380379341063813322432424, −6.81155066983315445218550683284, −5.82128168525755754297443293137, −5.21798747405709523541052327426, −4.94369911770872072111273594853, −3.85805870080380911684721325744, −2.91334044941297982903056670611, −2.22871318027998081417226803477, −1.14384835958507825698585314815, 0,
1.14384835958507825698585314815, 2.22871318027998081417226803477, 2.91334044941297982903056670611, 3.85805870080380911684721325744, 4.94369911770872072111273594853, 5.21798747405709523541052327426, 5.82128168525755754297443293137, 6.81155066983315445218550683284, 7.25556380379341063813322432424