| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 24·7-s + 8·8-s + 10·10-s + 13·13-s + 48·14-s + 16·16-s − 50·17-s + 28·19-s + 20·20-s + 208·23-s + 25·25-s + 26·26-s + 96·28-s − 190·29-s + 248·31-s + 32·32-s − 100·34-s + 120·35-s − 186·37-s + 56·38-s + 40·40-s + 194·41-s + 348·43-s + 416·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.29·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.916·14-s + 1/4·16-s − 0.713·17-s + 0.338·19-s + 0.223·20-s + 1.88·23-s + 1/5·25-s + 0.196·26-s + 0.647·28-s − 1.21·29-s + 1.43·31-s + 0.176·32-s − 0.504·34-s + 0.579·35-s − 0.826·37-s + 0.239·38-s + 0.158·40-s + 0.738·41-s + 1.23·43-s + 1.33·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.745967648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.745967648\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 208 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 186 T + p^{3} T^{2} \) |
| 41 | \( 1 - 194 T + p^{3} T^{2} \) |
| 43 | \( 1 - 348 T + p^{3} T^{2} \) |
| 47 | \( 1 + 260 T + p^{3} T^{2} \) |
| 53 | \( 1 + 462 T + p^{3} T^{2} \) |
| 59 | \( 1 - 520 T + p^{3} T^{2} \) |
| 61 | \( 1 + 506 T + p^{3} T^{2} \) |
| 67 | \( 1 - 772 T + p^{3} T^{2} \) |
| 71 | \( 1 + 780 T + p^{3} T^{2} \) |
| 73 | \( 1 + 62 T + p^{3} T^{2} \) |
| 79 | \( 1 - 736 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1464 T + p^{3} T^{2} \) |
| 89 | \( 1 + 406 T + p^{3} T^{2} \) |
| 97 | \( 1 - 922 T + p^{3} T^{2} \) |
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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
−9.328402046290813444719093413984, −8.569924596801474417032500313160, −7.64158110121363506229866884594, −6.85032911527224702079071401382, −5.86875977526381835288147663015, −5.02097307598911034101737757298, −4.43141370071466348854413812745, −3.16632770506712149218670739186, −2.08834936559310449870325219048, −1.09995323440738709143065472318,
1.09995323440738709143065472318, 2.08834936559310449870325219048, 3.16632770506712149218670739186, 4.43141370071466348854413812745, 5.02097307598911034101737757298, 5.86875977526381835288147663015, 6.85032911527224702079071401382, 7.64158110121363506229866884594, 8.569924596801474417032500313160, 9.328402046290813444719093413984
