| L(s) = 1 | + 4-s − 5-s − 9-s + 16-s − 2·19-s − 20-s + 25-s − 31-s − 36-s + 2·41-s + 45-s + 49-s + 2·59-s + 64-s − 2·71-s − 2·76-s − 80-s + 81-s + 2·95-s + 100-s − 2·101-s − 2·109-s + ⋯ |
| L(s) = 1 | + 4-s − 5-s − 9-s + 16-s − 2·19-s − 20-s + 25-s − 31-s − 36-s + 2·41-s + 45-s + 49-s + 2·59-s + 64-s − 2·71-s − 2·76-s − 80-s + 81-s + 2·95-s + 100-s − 2·101-s − 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6508459289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6508459289\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.92233217732296332972071056813, −12.06447657650733376192840104955, −11.17908071778950014910667656018, −10.60304083766435038603704405493, −8.861280814833111961532676385837, −7.954415932045116536338012755008, −6.88311713143119363974484421210, −5.75592926400217348902935065499, −4.01598014445173499465398203594, −2.56526779280730246524432404194,
2.56526779280730246524432404194, 4.01598014445173499465398203594, 5.75592926400217348902935065499, 6.88311713143119363974484421210, 7.954415932045116536338012755008, 8.861280814833111961532676385837, 10.60304083766435038603704405493, 11.17908071778950014910667656018, 12.06447657650733376192840104955, 12.92233217732296332972071056813
