| L(s) = 1 | − 6·3-s + 6·5-s + 14·7-s + 27·9-s + 26·11-s − 96·13-s − 36·15-s + 78·17-s + 40·19-s − 84·21-s − 22·23-s + 114·25-s − 108·27-s − 204·29-s − 96·31-s − 156·33-s + 84·35-s − 504·37-s + 576·39-s − 102·41-s + 296·43-s + 162·45-s + 780·47-s + 147·49-s − 468·51-s − 192·53-s + 156·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.536·5-s + 0.755·7-s + 9-s + 0.712·11-s − 2.04·13-s − 0.619·15-s + 1.11·17-s + 0.482·19-s − 0.872·21-s − 0.199·23-s + 0.911·25-s − 0.769·27-s − 1.30·29-s − 0.556·31-s − 0.822·33-s + 0.405·35-s − 2.23·37-s + 2.36·39-s − 0.388·41-s + 1.04·43-s + 0.536·45-s + 2.42·47-s + 3/7·49-s − 1.28·51-s − 0.497·53-s + 0.382·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.349690896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.349690896\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 6 T - 78 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 26 T + 2494 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 96 T + 5350 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 78 T + 8314 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 22 T + 16030 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 96 T - 24386 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 504 T + 163462 T^{2} + 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 102 T + 99666 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 296 T + 94646 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 780 T + 326046 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 192 T + 197782 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 212 T + 355942 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 100 T + 370190 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 212 T + 611414 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 534 T + 746334 T^{2} - 534 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1128 T + 1062430 T^{2} - 1128 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 468 T - 92834 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 824 T + 536870 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2118 T + 2285746 T^{2} + 2118 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 400 T + 214046 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.529318730113229696698132021893, −9.210949334217546792514199524444, −8.685926108470335337879632392499, −8.226566167661002014885380645478, −7.56701942131281943477060632010, −7.30842800767567648022968944062, −7.05299174035324700957338075233, −6.71747071058154931673161447108, −5.82970769088165133133417420616, −5.72372483680340028181148966939, −5.18796225017864545258680809254, −5.11265456856651801589645306924, −4.43729496964940819639734107947, −3.99741477738741610552388225265, −3.41151339734414707905532399432, −2.74683453230158137437032077880, −1.93891879003184181229858612331, −1.79952437172019671701933941579, −0.903523985323466834023089567552, −0.46663864824742395026595539611,
0.46663864824742395026595539611, 0.903523985323466834023089567552, 1.79952437172019671701933941579, 1.93891879003184181229858612331, 2.74683453230158137437032077880, 3.41151339734414707905532399432, 3.99741477738741610552388225265, 4.43729496964940819639734107947, 5.11265456856651801589645306924, 5.18796225017864545258680809254, 5.72372483680340028181148966939, 5.82970769088165133133417420616, 6.71747071058154931673161447108, 7.05299174035324700957338075233, 7.30842800767567648022968944062, 7.56701942131281943477060632010, 8.226566167661002014885380645478, 8.685926108470335337879632392499, 9.210949334217546792514199524444, 9.529318730113229696698132021893
