| L(s) = 1 | − 3.06e8·3-s + 2.74e11·4-s + 3.60e13·5-s − 1.25e18·9-s − 6.11e19·11-s − 8.43e19·12-s − 1.10e22·15-s + 7.55e22·16-s + 9.91e24·20-s + 9.84e25·23-s + 9.36e26·25-s + 8.00e26·27-s − 1.75e28·31-s + 1.87e28·33-s − 3.45e29·36-s + 3.26e29·37-s − 1.68e31·44-s − 4.53e31·45-s + 1.11e32·47-s − 2.31e31·48-s + 1.29e32·49-s + 1.04e33·53-s − 2.20e33·55-s − 3.57e33·59-s − 3.04e33·60-s + 2.07e34·64-s + 9.45e34·67-s + ⋯ |
| L(s) = 1 | − 0.264·3-s + 4-s + 1.89·5-s − 0.930·9-s − 11-s − 0.264·12-s − 0.499·15-s + 16-s + 1.89·20-s + 1.31·23-s + 2.57·25-s + 0.509·27-s − 0.808·31-s + 0.264·33-s − 0.930·36-s + 0.522·37-s − 44-s − 1.75·45-s + 1.88·47-s − 0.264·48-s + 49-s + 1.81·53-s − 1.89·55-s − 0.806·59-s − 0.499·60-s + 64-s + 1.90·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{39}{2})\) |
\(\approx\) |
\(3.703021340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.703021340\) |
| \(L(20)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + p^{19} T \) |
| good | 2 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 3 | \( 1 + 306986045 T + p^{38} T^{2} \) |
| 5 | \( 1 - 36059071233599 T + p^{38} T^{2} \) |
| 7 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 13 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 17 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 19 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 23 | \( 1 - \)\(98\!\cdots\!15\)\( T + p^{38} T^{2} \) |
| 29 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 31 | \( 1 + \)\(17\!\cdots\!17\)\( T + p^{38} T^{2} \) |
| 37 | \( 1 - \)\(32\!\cdots\!75\)\( T + p^{38} T^{2} \) |
| 41 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 43 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 47 | \( 1 - \)\(11\!\cdots\!50\)\( T + p^{38} T^{2} \) |
| 53 | \( 1 - \)\(10\!\cdots\!70\)\( T + p^{38} T^{2} \) |
| 59 | \( 1 + \)\(35\!\cdots\!53\)\( T + p^{38} T^{2} \) |
| 61 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 67 | \( 1 - \)\(94\!\cdots\!15\)\( T + p^{38} T^{2} \) |
| 71 | \( 1 + \)\(25\!\cdots\!13\)\( T + p^{38} T^{2} \) |
| 73 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 79 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 83 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 89 | \( 1 + \)\(21\!\cdots\!57\)\( T + p^{38} T^{2} \) |
| 97 | \( 1 - \)\(32\!\cdots\!55\)\( T + p^{38} T^{2} \) |
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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
−12.66337924460358267253678101649, −11.06231080913292540419861930004, −10.25265046799837070377896660249, −8.876223153961774212876126719389, −7.12016326003690952916473862235, −5.88923106015282402142329508079, −5.35085256083037358423743965067, −2.83940514094681936372407958327, −2.22969264020810138342894947236, −0.951527292231815159505027860858,
0.951527292231815159505027860858, 2.22969264020810138342894947236, 2.83940514094681936372407958327, 5.35085256083037358423743965067, 5.88923106015282402142329508079, 7.12016326003690952916473862235, 8.876223153961774212876126719389, 10.25265046799837070377896660249, 11.06231080913292540419861930004, 12.66337924460358267253678101649
