| L(s) = 1 | − 4·7-s + 2·9-s − 2·11-s + 8·13-s − 8·17-s + 4·29-s + 4·37-s + 12·41-s − 12·43-s − 2·49-s − 12·53-s + 8·59-s + 4·61-s − 8·63-s + 8·67-s + 8·73-s + 8·77-s − 8·79-s − 5·81-s − 12·83-s − 4·89-s − 32·91-s + 4·97-s − 4·99-s − 4·101-s + 8·103-s + 4·107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2/3·9-s − 0.603·11-s + 2.21·13-s − 1.94·17-s + 0.742·29-s + 0.657·37-s + 1.87·41-s − 1.82·43-s − 2/7·49-s − 1.64·53-s + 1.04·59-s + 0.512·61-s − 1.00·63-s + 0.977·67-s + 0.936·73-s + 0.911·77-s − 0.900·79-s − 5/9·81-s − 1.31·83-s − 0.423·89-s − 3.35·91-s + 0.406·97-s − 0.402·99-s − 0.398·101-s + 0.788·103-s + 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551661902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551661902\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.428917666442485204721965496020, −8.307323129530721521352898678046, −7.945100317910340946250159784051, −7.39668417147985976957974068277, −6.86162975841326349451970645629, −6.62129709077873261109247372357, −6.47442030103868018514899359095, −6.19196640897290924996541778752, −5.59080505299226006923794607029, −5.39851054544361617128403997444, −4.65178540046334925296204728728, −4.28083637001601474954894072663, −4.12401290521254449406907223166, −3.54437219555391260595094887917, −3.05568414663544724309254530475, −2.92045989125510829866317990895, −2.15662418413773366402103920806, −1.72686802855194500088937955603, −1.04628851899055958772267805654, −0.38995547894975355180916965397,
0.38995547894975355180916965397, 1.04628851899055958772267805654, 1.72686802855194500088937955603, 2.15662418413773366402103920806, 2.92045989125510829866317990895, 3.05568414663544724309254530475, 3.54437219555391260595094887917, 4.12401290521254449406907223166, 4.28083637001601474954894072663, 4.65178540046334925296204728728, 5.39851054544361617128403997444, 5.59080505299226006923794607029, 6.19196640897290924996541778752, 6.47442030103868018514899359095, 6.62129709077873261109247372357, 6.86162975841326349451970645629, 7.39668417147985976957974068277, 7.945100317910340946250159784051, 8.307323129530721521352898678046, 8.428917666442485204721965496020
