| L(s) = 1 | − 1.04e10·3-s + 4.39e12·4-s + 1.46e17·7-s + 1.09e20·9-s − 4.60e22·12-s − 3.57e23·13-s + 1.93e25·16-s − 1.65e26·19-s − 1.52e27·21-s + 2.27e29·25-s − 1.14e30·27-s + 6.43e29·28-s + 4.00e31·31-s + 4.81e32·36-s + 1.62e33·37-s + 3.73e33·39-s + 3.01e34·43-s − 2.02e35·48-s − 2.90e35·49-s − 1.57e36·52-s + 1.72e36·57-s + 5.58e37·61-s + 1.60e37·63-s + 8.50e37·64-s − 3.97e38·67-s − 1.16e39·73-s − 2.37e39·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 0.261·7-s + 9-s − 12-s − 1.44·13-s + 16-s − 0.231·19-s − 0.261·21-s + 25-s − 27-s + 0.261·28-s + 1.92·31-s + 36-s + 1.90·37-s + 1.44·39-s + 1.50·43-s − 48-s − 0.931·49-s − 1.44·52-s + 0.231·57-s + 1.80·61-s + 0.261·63-s + 64-s − 1.78·67-s − 0.866·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{43}{2})\) |
\(\approx\) |
\(1.846569382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.846569382\) |
| \(L(22)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{21} T \) |
| good | 2 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 5 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 7 | \( 1 - 146246101081752386 T + p^{42} T^{2} \) |
| 11 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 13 | \( 1 + \)\(35\!\cdots\!26\)\( T + p^{42} T^{2} \) |
| 17 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 19 | \( 1 + \)\(16\!\cdots\!62\)\( T + p^{42} T^{2} \) |
| 23 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 29 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 31 | \( 1 - \)\(40\!\cdots\!62\)\( T + p^{42} T^{2} \) |
| 37 | \( 1 - \)\(16\!\cdots\!26\)\( T + p^{42} T^{2} \) |
| 41 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 43 | \( 1 - \)\(30\!\cdots\!14\)\( T + p^{42} T^{2} \) |
| 47 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 53 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 59 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 61 | \( 1 - \)\(55\!\cdots\!22\)\( T + p^{42} T^{2} \) |
| 67 | \( 1 + \)\(39\!\cdots\!34\)\( T + p^{42} T^{2} \) |
| 71 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 73 | \( 1 + \)\(11\!\cdots\!46\)\( T + p^{42} T^{2} \) |
| 79 | \( 1 - \)\(14\!\cdots\!58\)\( T + p^{42} T^{2} \) |
| 83 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 89 | \( ( 1 - p^{21} T )( 1 + p^{21} T ) \) |
| 97 | \( 1 - \)\(22\!\cdots\!06\)\( T + p^{42} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.35827235700219296252281937246, −14.91787482411171572568475666686, −12.51198673522446325747556593263, −11.39928953835505891389508257875, −10.08796942496801273191380363359, −7.51929882772980564874026072286, −6.25969698317905202930938137396, −4.72969493735444101595200033906, −2.48208481356831408569546049547, −0.890144053637415040605839661333,
0.890144053637415040605839661333, 2.48208481356831408569546049547, 4.72969493735444101595200033906, 6.25969698317905202930938137396, 7.51929882772980564874026072286, 10.08796942496801273191380363359, 11.39928953835505891389508257875, 12.51198673522446325747556593263, 14.91787482411171572568475666686, 16.35827235700219296252281937246
