| L(s) = 1 | − 3-s − 2·9-s − 5·11-s + 13-s + 3·17-s + 6·19-s + 6·23-s + 5·27-s − 9·29-s + 5·33-s − 6·37-s − 39-s − 8·41-s − 6·43-s + 3·47-s − 3·51-s + 12·53-s − 6·57-s − 8·59-s + 4·61-s + 4·67-s − 6·69-s + 8·71-s + 10·73-s − 3·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.50·11-s + 0.277·13-s + 0.727·17-s + 1.37·19-s + 1.25·23-s + 0.962·27-s − 1.67·29-s + 0.870·33-s − 0.986·37-s − 0.160·39-s − 1.24·41-s − 0.914·43-s + 0.437·47-s − 0.420·51-s + 1.64·53-s − 0.794·57-s − 1.04·59-s + 0.512·61-s + 0.488·67-s − 0.722·69-s + 0.949·71-s + 1.17·73-s − 0.337·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−7.35506837323567972485067911086, −6.68703256181804606281147125839, −5.69124619217865701050329595606, −5.29384395404242793953990195152, −4.98809239944162573811126658586, −3.58039186109375468513509715851, −3.14114445601193552512668015535, −2.21471885119434180517790562935, −1.03834075997729573077765602229, 0,
1.03834075997729573077765602229, 2.21471885119434180517790562935, 3.14114445601193552512668015535, 3.58039186109375468513509715851, 4.98809239944162573811126658586, 5.29384395404242793953990195152, 5.69124619217865701050329595606, 6.68703256181804606281147125839, 7.35506837323567972485067911086
