L(s) = 1 | − 3-s + 3·5-s + 9-s + 3·11-s − 4·13-s − 3·15-s + 4·19-s + 4·25-s − 27-s − 9·29-s − 31-s − 3·33-s − 8·37-s + 4·39-s − 10·43-s + 3·45-s − 6·47-s + 3·53-s + 9·55-s − 4·57-s − 3·59-s − 10·61-s − 12·65-s − 10·67-s + 6·71-s − 2·73-s − 4·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 0.774·15-s + 0.917·19-s + 4/5·25-s − 0.192·27-s − 1.67·29-s − 0.179·31-s − 0.522·33-s − 1.31·37-s + 0.640·39-s − 1.52·43-s + 0.447·45-s − 0.875·47-s + 0.412·53-s + 1.21·55-s − 0.529·57-s − 0.390·59-s − 1.28·61-s − 1.48·65-s − 1.22·67-s + 0.712·71-s − 0.234·73-s − 0.461·75-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 | \( 1 - 3 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.15519974638240044232312116987, −6.65576177390318793233653232494, −5.93446977533954922284827924719, −5.32884723411701262067755392974, −4.87813131256828474700563618671, −3.84147733870847156766563133225, −2.99286013764661084506554020252, −1.91025704775030446334285168277, −1.45009779468465133708117857748, 0,
1.45009779468465133708117857748, 1.91025704775030446334285168277, 2.99286013764661084506554020252, 3.84147733870847156766563133225, 4.87813131256828474700563618671, 5.32884723411701262067755392974, 5.93446977533954922284827924719, 6.65576177390318793233653232494, 7.15519974638240044232312116987