L(s) = 1 | − 5.24e5·2-s + 4.01e8·3-s + 2.74e11·4-s − 2.10e14·6-s − 1.44e17·8-s − 1.18e18·9-s − 6.28e19·11-s + 1.10e20·12-s + 7.55e22·16-s − 4.30e23·17-s + 6.23e23·18-s + 3.20e24·19-s + 3.29e25·22-s − 5.79e25·24-s + 3.63e26·25-s − 1.02e27·27-s − 3.96e28·32-s − 2.52e28·33-s + 2.25e29·34-s − 3.26e29·36-s − 1.67e30·38-s − 8.34e30·41-s − 7.54e29·43-s − 1.72e31·44-s + 3.03e31·48-s + 1.29e32·49-s − 1.90e32·50-s + ⋯ |
L(s) = 1 | − 2-s + 0.345·3-s + 4-s − 0.345·6-s − 8-s − 0.880·9-s − 1.02·11-s + 0.345·12-s + 16-s − 1.79·17-s + 0.880·18-s + 1.61·19-s + 1.02·22-s − 0.345·24-s + 25-s − 0.650·27-s − 32-s − 0.355·33-s + 1.79·34-s − 0.880·36-s − 1.61·38-s − 1.89·41-s − 0.0694·43-s − 1.02·44-s + 0.345·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{39}{2})\) |
\(\approx\) |
\(0.9557595616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9557595616\) |
\(L(20)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{19} T \) |
good | 3 | \( 1 - 401778862 T + p^{38} T^{2} \) |
| 5 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 7 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 11 | \( 1 + 62820434121898204066 T + p^{38} T^{2} \) |
| 13 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 17 | \( 1 + \)\(43\!\cdots\!62\)\( T + p^{38} T^{2} \) |
| 19 | \( 1 - \)\(32\!\cdots\!86\)\( T + p^{38} T^{2} \) |
| 23 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 29 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 31 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 37 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 41 | \( 1 + \)\(83\!\cdots\!06\)\( T + p^{38} T^{2} \) |
| 43 | \( 1 + \)\(75\!\cdots\!14\)\( T + p^{38} T^{2} \) |
| 47 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 53 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 59 | \( 1 - \)\(79\!\cdots\!78\)\( T + p^{38} T^{2} \) |
| 61 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 67 | \( 1 + \)\(27\!\cdots\!02\)\( T + p^{38} T^{2} \) |
| 71 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 73 | \( 1 - \)\(12\!\cdots\!22\)\( T + p^{38} T^{2} \) |
| 79 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 83 | \( 1 - \)\(54\!\cdots\!22\)\( T + p^{38} T^{2} \) |
| 89 | \( 1 - \)\(11\!\cdots\!06\)\( T + p^{38} T^{2} \) |
| 97 | \( 1 + \)\(20\!\cdots\!66\)\( T + p^{38} T^{2} \) |
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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
−13.47524227353262008278783488171, −11.71858292827228083181890652308, −10.59243965397737344699975408883, −9.160242986209513506976465101791, −8.181694470107542906511450175519, −6.88088236187922687119156161451, −5.32164581086886974725893267283, −3.13348545554262342282045911132, −2.16344320293592441164032753381, −0.54979876145787308837673150238,
0.54979876145787308837673150238, 2.16344320293592441164032753381, 3.13348545554262342282045911132, 5.32164581086886974725893267283, 6.88088236187922687119156161451, 8.181694470107542906511450175519, 9.160242986209513506976465101791, 10.59243965397737344699975408883, 11.71858292827228083181890652308, 13.47524227353262008278783488171