| L(s) = 1 | + 2.04e3·2-s + 2.09e6·4-s + 4.15e7·5-s + 8.51e10·10-s − 1.18e12·13-s − 4.39e12·16-s + 2.20e13·17-s + 8.71e13·20-s + 1.25e15·25-s − 2.42e15·26-s − 9.00e15·32-s + 4.51e16·34-s + 7.41e16·37-s − 3.43e17·41-s + 2.56e18·50-s − 2.48e18·52-s + 1.22e18·53-s − 9.74e18·61-s − 9.22e18·64-s − 4.92e19·65-s + 4.62e19·68-s − 9.28e19·73-s + 1.51e20·74-s − 1.82e20·80-s − 1.09e20·81-s − 7.03e20·82-s + 9.16e20·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.90·5-s + 2.69·10-s − 2.38·13-s − 16-s + 2.65·17-s + 1.90·20-s + 2.62·25-s − 3.37·26-s − 1.41·32-s + 3.75·34-s + 2.53·37-s − 3.99·41-s + 3.71·50-s − 2.38·52-s + 0.960·53-s − 1.74·61-s − 64-s − 4.54·65-s + 2.65·68-s − 2.52·73-s + 3.58·74-s − 1.90·80-s − 81-s − 5.65·82-s + 5.05·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(8.764638911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.764638911\) |
| \(L(\frac{23}{2})\) |
|
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 | $C_2$ | \( 1 - p^{11} T + p^{21} T^{2} \) |
| 5 | $C_2$ | \( 1 - 41567116 T + p^{21} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{21} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 215310526796 T + p^{21} T^{2} )( 1 + 970515066006 T + p^{21} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 15095904884798 T + p^{21} T^{2} )( 1 - 6962857988008 T + p^{21} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{21} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 617267444622790 T + p^{21} T^{2} )( 1 + 617267444622790 T + p^{21} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{21} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 55407939890055588 T + p^{21} T^{2} )( 1 - 18764002233154798 T + p^{21} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 171803734370535208 T + p^{21} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2305059469250980414 T + p^{21} T^{2} )( 1 + 1082059993196406796 T + p^{21} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{21} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4870155553439314788 T + p^{21} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{21} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 23086983136439707216 T + p^{21} T^{2} )( 1 + 69714495186591036006 T + p^{21} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{21} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{42} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - \)\(43\!\cdots\!90\)\( T + p^{21} T^{2} )( 1 + \)\(43\!\cdots\!90\)\( T + p^{21} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - \)\(86\!\cdots\!08\)\( T + p^{21} T^{2} )( 1 + \)\(11\!\cdots\!82\)\( T + p^{21} T^{2} ) \) |
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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
−14.25051829154295742300208400610, −13.15225718285945647327861840672, −12.81883027670999444867619603102, −11.98096205664058392547200978307, −11.82645292723587370390192362615, −10.36898500357573671081268595481, −9.874628633682215048967141593746, −9.703921862747379807658037635640, −8.671429215790447072881769416543, −7.51071154727426746494247431713, −6.98201000792338960397359852635, −5.93461488415315789998387284426, −5.74121815871819107705499377746, −4.92408700134245283962330955635, −4.68420358236925903270578846616, −3.19662625265798288652667475151, −2.97123128440317506990390353146, −2.13555863354284580448345249747, −1.55724061711642973077294908131, −0.56242199730680107753845031783,
0.56242199730680107753845031783, 1.55724061711642973077294908131, 2.13555863354284580448345249747, 2.97123128440317506990390353146, 3.19662625265798288652667475151, 4.68420358236925903270578846616, 4.92408700134245283962330955635, 5.74121815871819107705499377746, 5.93461488415315789998387284426, 6.98201000792338960397359852635, 7.51071154727426746494247431713, 8.671429215790447072881769416543, 9.703921862747379807658037635640, 9.874628633682215048967141593746, 10.36898500357573671081268595481, 11.82645292723587370390192362615, 11.98096205664058392547200978307, 12.81883027670999444867619603102, 13.15225718285945647327861840672, 14.25051829154295742300208400610
