| L(s) = 1 | + 4·3-s + 16·7-s − 11·9-s + 60·11-s + 86·13-s − 18·17-s − 44·19-s + 64·21-s − 48·23-s − 152·27-s + 186·29-s + 176·31-s + 240·33-s + 254·37-s + 344·39-s + 186·41-s − 100·43-s − 168·47-s − 87·49-s − 72·51-s − 498·53-s − 176·57-s + 252·59-s + 58·61-s − 176·63-s − 1.03e3·67-s − 192·69-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.863·7-s − 0.407·9-s + 1.64·11-s + 1.83·13-s − 0.256·17-s − 0.531·19-s + 0.665·21-s − 0.435·23-s − 1.08·27-s + 1.19·29-s + 1.01·31-s + 1.26·33-s + 1.12·37-s + 1.41·39-s + 0.708·41-s − 0.354·43-s − 0.521·47-s − 0.253·49-s − 0.197·51-s − 1.29·53-s − 0.408·57-s + 0.556·59-s + 0.121·61-s − 0.351·63-s − 1.88·67-s − 0.334·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.956599565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.956599565\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 18 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 186 T + p^{3} T^{2} \) |
| 43 | \( 1 + 100 T + p^{3} T^{2} \) |
| 47 | \( 1 + 168 T + p^{3} T^{2} \) |
| 53 | \( 1 + 498 T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 58 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 71 | \( 1 - 168 T + p^{3} T^{2} \) |
| 73 | \( 1 + 506 T + p^{3} T^{2} \) |
| 79 | \( 1 - 272 T + p^{3} T^{2} \) |
| 83 | \( 1 - 948 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1014 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−8.758828930504054777266266164948, −8.479042633818777473673211491960, −7.72566368646203341516972878494, −6.36128499145550891267018678784, −6.14084424044332491013225800274, −4.63776280931077446975326342680, −3.94833846521275419181462881539, −3.06498965916211964832585212667, −1.83928437341290200102179458621, −1.00940935772842030179320173557,
1.00940935772842030179320173557, 1.83928437341290200102179458621, 3.06498965916211964832585212667, 3.94833846521275419181462881539, 4.63776280931077446975326342680, 6.14084424044332491013225800274, 6.36128499145550891267018678784, 7.72566368646203341516972878494, 8.479042633818777473673211491960, 8.758828930504054777266266164948
