L(s) = 1 | − 2.14e9·2-s − 1.10e15·3-s + 4.61e18·4-s + 2.37e24·6-s − 9.90e27·8-s + 8.43e29·9-s − 2.19e32·11-s − 5.10e33·12-s + 2.12e37·16-s − 2.69e38·17-s − 1.81e39·18-s + 1.86e39·19-s + 4.70e41·22-s + 1.09e43·24-s + 2.16e43·25-s − 5.10e44·27-s − 4.56e46·32-s + 2.42e47·33-s + 5.79e47·34-s + 3.88e48·36-s − 4.00e48·38-s − 7.48e49·41-s + 8.13e50·43-s − 1.01e51·44-s − 2.35e52·48-s + 2.48e52·49-s − 4.65e52·50-s + ⋯ |
L(s) = 1 | − 2-s − 1.79·3-s + 4-s + 1.79·6-s − 8-s + 2.20·9-s − 1.14·11-s − 1.79·12-s + 16-s − 1.93·17-s − 2.20·18-s + 0.425·19-s + 1.14·22-s + 1.79·24-s + 25-s − 2.16·27-s − 32-s + 2.04·33-s + 1.93·34-s + 2.20·36-s − 0.425·38-s − 0.754·41-s + 1.87·43-s − 1.14·44-s − 1.79·48-s + 49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(63-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+31) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{63}{2})\) |
\(\approx\) |
\(0.1990784876\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1990784876\) |
\(L(32)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{31} T \) |
good | 3 | \( 1 + 1106589796438898 T + p^{62} T^{2} \) |
| 5 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 7 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 11 | \( 1 + \)\(21\!\cdots\!86\)\( T + p^{62} T^{2} \) |
| 13 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 17 | \( 1 + \)\(26\!\cdots\!02\)\( T + p^{62} T^{2} \) |
| 19 | \( 1 - \)\(18\!\cdots\!66\)\( T + p^{62} T^{2} \) |
| 23 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 29 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 31 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 37 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 41 | \( 1 + \)\(74\!\cdots\!46\)\( T + p^{62} T^{2} \) |
| 43 | \( 1 - \)\(81\!\cdots\!86\)\( T + p^{62} T^{2} \) |
| 47 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 53 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 59 | \( 1 + \)\(15\!\cdots\!82\)\( T + p^{62} T^{2} \) |
| 61 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 67 | \( 1 + \)\(58\!\cdots\!62\)\( T + p^{62} T^{2} \) |
| 71 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 73 | \( 1 + \)\(63\!\cdots\!58\)\( T + p^{62} T^{2} \) |
| 79 | \( ( 1 - p^{31} T )( 1 + p^{31} T ) \) |
| 83 | \( 1 + \)\(60\!\cdots\!58\)\( T + p^{62} T^{2} \) |
| 89 | \( 1 - \)\(53\!\cdots\!46\)\( T + p^{62} T^{2} \) |
| 97 | \( 1 - \)\(21\!\cdots\!94\)\( T + p^{62} T^{2} \) |
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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
−10.91704419162737643313929990188, −10.48418080713287983232278238675, −9.048658263296277425106830628684, −7.49307030262866447039512765204, −6.58359078375601761605324348140, −5.62199213909310443551481879697, −4.53507258093847892189838992300, −2.58820295985976050036845333167, −1.32086912849988127243100050645, −0.25980509838338912446674407737,
0.25980509838338912446674407737, 1.32086912849988127243100050645, 2.58820295985976050036845333167, 4.53507258093847892189838992300, 5.62199213909310443551481879697, 6.58359078375601761605324348140, 7.49307030262866447039512765204, 9.048658263296277425106830628684, 10.48418080713287983232278238675, 10.91704419162737643313929990188