| L(s) = 1 | + 4·3-s − 4·7-s + 8·9-s + 16·11-s + 6·13-s + 14·17-s − 16·21-s + 4·23-s − 25·25-s + 36·27-s − 104·31-s + 64·33-s − 6·37-s + 24·39-s − 16·41-s + 84·43-s + 36·47-s + 8·49-s + 56·51-s + 106·53-s − 96·61-s − 32·63-s − 124·67-s + 16·69-s + 56·71-s − 94·73-s − 100·75-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 4/7·7-s + 8/9·9-s + 1.45·11-s + 6/13·13-s + 0.823·17-s − 0.761·21-s + 4/23·23-s − 25-s + 4/3·27-s − 3.35·31-s + 1.93·33-s − 0.162·37-s + 8/13·39-s − 0.390·41-s + 1.95·43-s + 0.765·47-s + 8/49·49-s + 1.09·51-s + 2·53-s − 1.57·61-s − 0.507·63-s − 1.85·67-s + 0.231·69-s + 0.788·71-s − 1.28·73-s − 4/3·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.281966687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.281966687\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )( 1 + 16 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 124 T + 7688 T^{2} + 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−14.49993930591100624662516488587, −13.86044914999809610449259555471, −13.55539247026847928857388913450, −12.88772438575604322004032726204, −12.21262680069244875304099998371, −12.00391884968522702685756130805, −10.94941113218402555256988984869, −10.60743643636833107945835355572, −9.688183020099762016018134082624, −9.188745117235612226752961263905, −8.983568593412179507645337916561, −8.295764060792530018351770838178, −7.34712988900325271059569299119, −7.17546497060046303972154404160, −6.10199289521861824782559450555, −5.53305507942382811346062691126, −4.06184677211170776568625861228, −3.74710321636200446913837047237, −2.82748992707805876928492998427, −1.55983736865652693073481966028,
1.55983736865652693073481966028, 2.82748992707805876928492998427, 3.74710321636200446913837047237, 4.06184677211170776568625861228, 5.53305507942382811346062691126, 6.10199289521861824782559450555, 7.17546497060046303972154404160, 7.34712988900325271059569299119, 8.295764060792530018351770838178, 8.983568593412179507645337916561, 9.188745117235612226752961263905, 9.688183020099762016018134082624, 10.60743643636833107945835355572, 10.94941113218402555256988984869, 12.00391884968522702685756130805, 12.21262680069244875304099998371, 12.88772438575604322004032726204, 13.55539247026847928857388913450, 13.86044914999809610449259555471, 14.49993930591100624662516488587
