| L(s) = 1 | + 3-s + 7-s + 9-s − 2·17-s − 4·19-s + 21-s − 5·25-s + 27-s − 4·29-s − 8·31-s + 4·37-s − 10·41-s + 49-s − 2·51-s + 4·53-s − 4·57-s − 4·59-s − 8·61-s + 63-s − 8·67-s − 6·73-s − 5·75-s + 8·79-s + 81-s − 4·83-s − 4·87-s − 6·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 25-s + 0.192·27-s − 0.742·29-s − 1.43·31-s + 0.657·37-s − 1.56·41-s + 1/7·49-s − 0.280·51-s + 0.549·53-s − 0.529·57-s − 0.520·59-s − 1.02·61-s + 0.125·63-s − 0.977·67-s − 0.702·73-s − 0.577·75-s + 0.900·79-s + 1/9·81-s − 0.439·83-s − 0.428·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.77094226171602071087950256092, −7.30801589945421700796209963363, −6.40679105319414166076336493937, −5.68926356243390349634082409311, −4.77675541154972176426031054405, −4.05849895593562737047846516402, −3.32789993302974585994067643384, −2.24026872413168081114069147504, −1.62537570496439834117582978709, 0,
1.62537570496439834117582978709, 2.24026872413168081114069147504, 3.32789993302974585994067643384, 4.05849895593562737047846516402, 4.77675541154972176426031054405, 5.68926356243390349634082409311, 6.40679105319414166076336493937, 7.30801589945421700796209963363, 7.77094226171602071087950256092
