| L(s) = 1 | − 2·3-s − 5-s − 2·7-s + 9-s − 4·11-s − 6·13-s + 2·15-s + 2·17-s + 8·19-s + 4·21-s − 6·23-s + 25-s + 4·27-s − 2·29-s + 4·31-s + 8·33-s + 2·35-s + 2·37-s + 12·39-s − 10·41-s − 2·43-s − 45-s − 2·47-s − 3·49-s − 4·51-s + 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.516·15-s + 0.485·17-s + 1.83·19-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s + 1.39·33-s + 0.338·35-s + 0.328·37-s + 1.92·39-s − 1.56·41-s − 0.304·43-s − 0.149·45-s − 0.291·47-s − 3/7·49-s − 0.560·51-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.08463452386231960892776929550, −11.70642323602707878068391757381, −10.29364010457156446481778884088, −9.759510932768572667567645779831, −7.967338729692305876626971946668, −7.05269675268361745315034445294, −5.70243463207358237884993422463, −4.88847555485350104575427844162, −3.01064345970677263239408332098, 0,
3.01064345970677263239408332098, 4.88847555485350104575427844162, 5.70243463207358237884993422463, 7.05269675268361745315034445294, 7.967338729692305876626971946668, 9.759510932768572667567645779831, 10.29364010457156446481778884088, 11.70642323602707878068391757381, 12.08463452386231960892776929550
