| L(s) = 1 | + 7-s − 4·11-s − 6·13-s + 23-s − 2·29-s + 2·31-s − 2·37-s + 6·41-s − 4·43-s − 2·47-s + 49-s + 14·53-s − 14·59-s + 12·61-s + 4·67-s + 2·73-s − 4·77-s − 8·79-s − 4·83-s + 16·89-s − 6·91-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.20·11-s − 1.66·13-s + 0.208·23-s − 0.371·29-s + 0.359·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.92·53-s − 1.82·59-s + 1.53·61-s + 0.488·67-s + 0.234·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s + 1.69·89-s − 0.628·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048907423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048907423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.65921178928641, −12.22970436574344, −11.94764392797834, −11.21716034901071, −10.99475389859249, −10.31824291475581, −9.968976153548739, −9.691481113218201, −8.936627668971565, −8.587110621289906, −7.971753357856179, −7.538444825130087, −7.283634706306049, −6.711147486275735, −6.039691414919562, −5.488960486544926, −5.029307765002042, −4.777870139533465, −4.106559699209955, −3.490979943557369, −2.735200126507364, −2.453682850885533, −1.922270068452004, −1.064211497058905, −0.2884694646670270,
0.2884694646670270, 1.064211497058905, 1.922270068452004, 2.453682850885533, 2.735200126507364, 3.490979943557369, 4.106559699209955, 4.777870139533465, 5.029307765002042, 5.488960486544926, 6.039691414919562, 6.711147486275735, 7.283634706306049, 7.538444825130087, 7.971753357856179, 8.587110621289906, 8.936627668971565, 9.691481113218201, 9.968976153548739, 10.31824291475581, 10.99475389859249, 11.21716034901071, 11.94764392797834, 12.22970436574344, 12.65921178928641
