L(s) = 1 | + 4·3-s − 2·5-s + 6·9-s + 5·11-s − 8·15-s + 16·23-s − 7·25-s − 4·27-s + 8·31-s + 20·33-s + 20·37-s − 12·45-s − 2·47-s − 5·49-s − 8·53-s − 10·55-s + 12·59-s − 24·67-s + 64·69-s + 4·71-s − 28·75-s − 37·81-s + 24·89-s + 32·93-s − 16·97-s + 30·99-s − 12·103-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 0.894·5-s + 2·9-s + 1.50·11-s − 2.06·15-s + 3.33·23-s − 7/5·25-s − 0.769·27-s + 1.43·31-s + 3.48·33-s + 3.28·37-s − 1.78·45-s − 0.291·47-s − 5/7·49-s − 1.09·53-s − 1.34·55-s + 1.56·59-s − 2.93·67-s + 7.70·69-s + 0.474·71-s − 3.23·75-s − 4.11·81-s + 2.54·89-s + 3.31·93-s − 1.62·97-s + 3.01·99-s − 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 698896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.369608332\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.369608332\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.305060519445702337530152822868, −7.933665480997825519415597461102, −7.62429229152375892340957919668, −7.32424896802954806677465404753, −6.44587861158160350440717785391, −6.44323514092267885981017350010, −5.57209357785375076961912739494, −4.85244736948764917088509248105, −4.15537276955970084853193543262, −4.08026045073955845688679172688, −3.36880239235033917183403851835, −2.82511261070703297705797963037, −2.77013505033142815042798931153, −1.74404941065737830362193213595, −0.971870007004528213384011971816,
0.971870007004528213384011971816, 1.74404941065737830362193213595, 2.77013505033142815042798931153, 2.82511261070703297705797963037, 3.36880239235033917183403851835, 4.08026045073955845688679172688, 4.15537276955970084853193543262, 4.85244736948764917088509248105, 5.57209357785375076961912739494, 6.44323514092267885981017350010, 6.44587861158160350440717785391, 7.32424896802954806677465404753, 7.62429229152375892340957919668, 7.933665480997825519415597461102, 8.305060519445702337530152822868