| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s − 5·13-s − 15-s − 4·17-s − 4·19-s − 3·21-s − 2·23-s + 25-s + 27-s − 9·31-s + 3·35-s − 5·39-s − 10·41-s − 43-s − 45-s + 6·47-s + 2·49-s − 4·51-s + 2·53-s − 4·57-s + 4·59-s − 6·61-s − 3·63-s + 5·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s − 1.38·13-s − 0.258·15-s − 0.970·17-s − 0.917·19-s − 0.654·21-s − 0.417·23-s + 1/5·25-s + 0.192·27-s − 1.61·31-s + 0.507·35-s − 0.800·39-s − 1.56·41-s − 0.152·43-s − 0.149·45-s + 0.875·47-s + 2/7·49-s − 0.560·51-s + 0.274·53-s − 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.377·63-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.81410988244002, −12.50191840726225, −12.04732441272271, −11.51114580505302, −10.96551848234555, −10.47081155026676, −10.06167504187538, −9.569020480460807, −9.251176434926053, −8.627422165798235, −8.423412737134175, −7.640735712465519, −7.283697626671894, −6.865527770722934, −6.451011937646313, −5.897506729679470, −5.156559553965008, −4.800897065331813, −4.054862400216236, −3.805472880946885, −3.189625700269507, −2.613832802216411, −2.167794405390942, −1.637372394470418, −0.4746147834162538, 0,
0.4746147834162538, 1.637372394470418, 2.167794405390942, 2.613832802216411, 3.189625700269507, 3.805472880946885, 4.054862400216236, 4.800897065331813, 5.156559553965008, 5.897506729679470, 6.451011937646313, 6.865527770722934, 7.283697626671894, 7.640735712465519, 8.423412737134175, 8.627422165798235, 9.251176434926053, 9.569020480460807, 10.06167504187538, 10.47081155026676, 10.96551848234555, 11.51114580505302, 12.04732441272271, 12.50191840726225, 12.81410988244002
