L(s) = 1 | − 3-s + 2·7-s + 9-s − 2·13-s − 17-s + 4·19-s − 2·21-s − 6·23-s − 5·25-s − 27-s − 10·31-s − 8·37-s + 2·39-s + 6·41-s + 4·43-s + 12·47-s − 3·49-s + 51-s − 6·53-s − 4·57-s + 12·59-s − 8·61-s + 2·63-s + 4·67-s + 6·69-s + 6·71-s + 2·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.242·17-s + 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s − 1.79·31-s − 1.31·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s + 0.140·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s − 1.02·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s + 0.712·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.115102227964514339079390978935, −7.51476003917695097413986406817, −6.88843577578684808918223131504, −5.70964004788136228366901344496, −5.44237790561600373941790334940, −4.40279032004747662521271149929, −3.72111015541234406836981026227, −2.37227056817861814340285837209, −1.47870270507371948822204128989, 0,
1.47870270507371948822204128989, 2.37227056817861814340285837209, 3.72111015541234406836981026227, 4.40279032004747662521271149929, 5.44237790561600373941790334940, 5.70964004788136228366901344496, 6.88843577578684808918223131504, 7.51476003917695097413986406817, 8.115102227964514339079390978935