L(s) = 1 | + 3-s + 9-s + 6·11-s + 2·13-s − 4·19-s − 6·23-s − 5·25-s + 27-s − 6·29-s − 8·31-s + 6·33-s − 2·37-s + 2·39-s − 12·41-s + 4·43-s − 12·47-s + 6·53-s − 4·57-s − 10·61-s − 8·67-s − 6·69-s + 6·71-s + 10·73-s − 5·75-s − 4·79-s + 81-s − 12·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.917·19-s − 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s − 1.87·41-s + 0.609·43-s − 1.75·47-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.977·67-s − 0.722·69-s + 0.712·71-s + 1.17·73-s − 0.577·75-s − 0.450·79-s + 1/9·81-s − 1.31·83-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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.33978744208638478150447194311, −6.68018359677214897577974504311, −6.12118933536028124716817869873, −5.38931467471481918720560102508, −4.17509149223426232107219639461, −3.92899224204306789996334105668, −3.22673417481384035100368040671, −1.84987696198250663319067290149, −1.64262388040949769268493081981, 0,
1.64262388040949769268493081981, 1.84987696198250663319067290149, 3.22673417481384035100368040671, 3.92899224204306789996334105668, 4.17509149223426232107219639461, 5.38931467471481918720560102508, 6.12118933536028124716817869873, 6.68018359677214897577974504311, 7.33978744208638478150447194311