| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 2·11-s + 13-s + 15-s − 8·17-s − 2·19-s − 21-s + 23-s + 25-s + 5·27-s − 29-s − 7·31-s − 2·33-s − 35-s − 10·37-s − 39-s − 5·41-s + 8·43-s + 2·45-s − 3·47-s + 49-s + 8·51-s + 10·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 1.94·17-s − 0.458·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s − 1.25·31-s − 0.348·33-s − 0.169·35-s − 1.64·37-s − 0.160·39-s − 0.780·41-s + 1.21·43-s + 0.298·45-s − 0.437·47-s + 1/7·49-s + 1.12·51-s + 1.37·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.74531138204951, −14.35520835829450, −13.62691826826880, −13.27027386466110, −12.62781599130282, −11.99532843359250, −11.71595256260004, −11.01198083054475, −10.89560241380449, −10.36579913417116, −9.347172619734100, −8.966106256233728, −8.541867560613136, −8.053515369622446, −7.154134060810272, −6.750807684350595, −6.368237755636763, −5.434136272488648, −5.240172757824249, −4.345902872465344, −3.952827712574657, −3.262211419759875, −2.319589931995931, −1.832217127084239, −0.7508393128021893, 0,
0.7508393128021893, 1.832217127084239, 2.319589931995931, 3.262211419759875, 3.952827712574657, 4.345902872465344, 5.240172757824249, 5.434136272488648, 6.368237755636763, 6.750807684350595, 7.154134060810272, 8.053515369622446, 8.541867560613136, 8.966106256233728, 9.347172619734100, 10.36579913417116, 10.89560241380449, 11.01198083054475, 11.71595256260004, 11.99532843359250, 12.62781599130282, 13.27027386466110, 13.62691826826880, 14.35520835829450, 14.74531138204951
