| L(s) = 1 | + 2-s − 4-s − 3·8-s − 8·11-s − 16-s + 12·17-s + 8·19-s − 8·22-s − 6·25-s + 5·32-s + 12·34-s + 8·38-s − 4·41-s − 8·43-s + 8·44-s + 49-s − 6·50-s − 24·59-s + 7·64-s + 8·67-s − 12·68-s − 12·73-s − 8·76-s − 4·82-s + 24·83-s − 8·86-s + 24·88-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 2.41·11-s − 1/4·16-s + 2.91·17-s + 1.83·19-s − 1.70·22-s − 6/5·25-s + 0.883·32-s + 2.05·34-s + 1.29·38-s − 0.624·41-s − 1.21·43-s + 1.20·44-s + 1/7·49-s − 0.848·50-s − 3.12·59-s + 7/8·64-s + 0.977·67-s − 1.45·68-s − 1.40·73-s − 0.917·76-s − 0.441·82-s + 2.63·83-s − 0.862·86-s + 2.55·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.625547847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.625547847\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.042140676903253409137259496371, −8.229992153219430708166678625388, −7.81295430367747747098376616892, −7.70691183501011689426667743544, −7.32441936749744810989380005535, −6.14521652142582152085553341069, −5.94607665445287604157300008429, −5.34641745697560833577710874285, −5.02709416296405066539370067618, −4.80722452268725035282926398957, −3.62256199043071801163653681899, −3.22746506277116131252491518684, −3.05422074105458389777226041971, −1.93480858488759200478154145238, −0.69577924383102779401083388688,
0.69577924383102779401083388688, 1.93480858488759200478154145238, 3.05422074105458389777226041971, 3.22746506277116131252491518684, 3.62256199043071801163653681899, 4.80722452268725035282926398957, 5.02709416296405066539370067618, 5.34641745697560833577710874285, 5.94607665445287604157300008429, 6.14521652142582152085553341069, 7.32441936749744810989380005535, 7.70691183501011689426667743544, 7.81295430367747747098376616892, 8.229992153219430708166678625388, 9.042140676903253409137259496371
