| L(s) = 1 | + 2·3-s − 2·7-s − 6·11-s + 4·13-s − 2·17-s + 4·19-s − 4·21-s − 6·23-s + 2·25-s − 2·27-s − 2·31-s − 12·33-s + 16·37-s + 8·39-s − 12·41-s + 4·43-s − 8·49-s − 4·51-s + 12·53-s + 8·57-s + 12·59-s − 8·61-s + 16·67-s − 12·69-s − 6·71-s + 4·73-s + 4·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s − 1.80·11-s + 1.10·13-s − 0.485·17-s + 0.917·19-s − 0.872·21-s − 1.25·23-s + 2/5·25-s − 0.384·27-s − 0.359·31-s − 2.08·33-s + 2.63·37-s + 1.28·39-s − 1.87·41-s + 0.609·43-s − 8/7·49-s − 0.560·51-s + 1.64·53-s + 1.05·57-s + 1.56·59-s − 1.02·61-s + 1.95·67-s − 1.44·69-s − 0.712·71-s + 0.468·73-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9928162223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9928162223\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 180 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−14.98802756398736199380475449543, −14.54787209899213773355774481944, −13.71694179562516136643627390396, −13.64254954364421247043782466442, −12.89878498621886309848267626598, −12.75367942326120361788196272822, −11.53273234940845816447213415980, −11.32712987605443907568586060344, −10.27052246725503930045048556869, −10.07592054412312095329842479452, −9.263026414816488689752299613577, −8.646958824177432743855930500283, −8.106902717026493810841304916570, −7.70076766939605653065525734959, −6.73900062082143014830718293103, −5.93177045679069081961924231806, −5.24049826920759417255257161277, −4.02594556598422966888602223977, −3.12538302478095309559579133561, −2.47307964120050239999721494055,
2.47307964120050239999721494055, 3.12538302478095309559579133561, 4.02594556598422966888602223977, 5.24049826920759417255257161277, 5.93177045679069081961924231806, 6.73900062082143014830718293103, 7.70076766939605653065525734959, 8.106902717026493810841304916570, 8.646958824177432743855930500283, 9.263026414816488689752299613577, 10.07592054412312095329842479452, 10.27052246725503930045048556869, 11.32712987605443907568586060344, 11.53273234940845816447213415980, 12.75367942326120361788196272822, 12.89878498621886309848267626598, 13.64254954364421247043782466442, 13.71694179562516136643627390396, 14.54787209899213773355774481944, 14.98802756398736199380475449543
