| L(s) = 1 | − 260·3-s + 1.00e3·4-s + 4.79e4·9-s − 2.60e5·12-s − 1.65e5·13-s + 7.39e5·16-s − 3.90e6·25-s − 7.34e6·27-s + 1.99e7·31-s + 4.79e7·36-s + 4.29e7·39-s − 1.92e8·48-s + 8.07e7·49-s − 1.65e8·52-s + 4.78e8·64-s + 8.23e8·73-s + 1.01e9·75-s + 9.65e8·81-s − 5.18e9·93-s − 3.91e9·100-s − 7.34e9·108-s − 7.90e9·117-s − 4.71e9·121-s + 1.99e10·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.85·3-s + 1.95·4-s + 2.43·9-s − 3.62·12-s − 1.60·13-s + 2.82·16-s − 2·25-s − 2.65·27-s + 3.87·31-s + 4.75·36-s + 2.96·39-s − 5.23·48-s + 2·49-s − 3.13·52-s + 3.56·64-s + 3.39·73-s + 3.70·75-s + 2.49·81-s − 7.18·93-s − 3.91·100-s − 5.19·108-s − 3.90·117-s − 2·121-s + 7.57·124-s + 6.87·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.368047141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.368047141\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + 260 T + p^{9} T^{2} \) |
| 23 | $C_2$ | \( 1 + p^{9} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - 45 T + p^{9} T^{2} )( 1 + 45 T + p^{9} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 82510 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1792674 T + p^{9} T^{2} )( 1 + 1792674 T + p^{9} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9963272 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10772262 T + p^{9} T^{2} )( 1 + 10772262 T + p^{9} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 14839920 T + p^{9} T^{2} )( 1 + 14839920 T + p^{9} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 181891116 T + p^{9} T^{2} )( 1 + 181891116 T + p^{9} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 363669768 T + p^{9} T^{2} )( 1 + 363669768 T + p^{9} T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 411966650 T + p^{9} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{9} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{9} T^{2} )^{2} \) |
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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
−12.71259317746919945592686622179, −12.07113959299463746312442941844, −11.94028165537965936134945010134, −11.71809270744697919053949385749, −10.97444991155882792324650539138, −10.45243821549322894807319741706, −9.984601259754138788547641770772, −9.683440815437295668828979989711, −8.039748866261213181982488130659, −7.69705780548221605827569908784, −6.96812555482288185216199580963, −6.54329180931698529550110763156, −6.06621438620234255811380472319, −5.44410214236671421466094644884, −4.76017350870122186473888351452, −3.88386840631841173773104873785, −2.61165204712638531058080405535, −2.17619172917132475131712805294, −1.18060529906616511603153850066, −0.54912708692087913989799583468,
0.54912708692087913989799583468, 1.18060529906616511603153850066, 2.17619172917132475131712805294, 2.61165204712638531058080405535, 3.88386840631841173773104873785, 4.76017350870122186473888351452, 5.44410214236671421466094644884, 6.06621438620234255811380472319, 6.54329180931698529550110763156, 6.96812555482288185216199580963, 7.69705780548221605827569908784, 8.039748866261213181982488130659, 9.683440815437295668828979989711, 9.984601259754138788547641770772, 10.45243821549322894807319741706, 10.97444991155882792324650539138, 11.71809270744697919053949385749, 11.94028165537965936134945010134, 12.07113959299463746312442941844, 12.71259317746919945592686622179
