L(s) = 1 | − 2-s + 2.79·3-s + 4-s − 5-s − 2.79·6-s − 0.0686·7-s − 8-s + 4.80·9-s + 10-s − 3.78·11-s + 2.79·12-s + 5.08·13-s + 0.0686·14-s − 2.79·15-s + 16-s + 2.10·17-s − 4.80·18-s − 7.00·19-s − 20-s − 0.191·21-s + 3.78·22-s − 3.91·23-s − 2.79·24-s + 25-s − 5.08·26-s + 5.04·27-s − 0.0686·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.61·3-s + 0.5·4-s − 0.447·5-s − 1.14·6-s − 0.0259·7-s − 0.353·8-s + 1.60·9-s + 0.316·10-s − 1.14·11-s + 0.806·12-s + 1.40·13-s + 0.0183·14-s − 0.721·15-s + 0.250·16-s + 0.510·17-s − 1.13·18-s − 1.60·19-s − 0.223·20-s − 0.0418·21-s + 0.808·22-s − 0.816·23-s − 0.570·24-s + 0.200·25-s − 0.996·26-s + 0.970·27-s − 0.0129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 7 | \( 1 + 0.0686T + 7T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 + 5.45T + 31T^{2} \) |
| 37 | \( 1 - 4.23T + 37T^{2} \) |
| 41 | \( 1 - 0.618T + 41T^{2} \) |
| 43 | \( 1 + 8.27T + 43T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 2.77T + 59T^{2} \) |
| 61 | \( 1 + 3.00T + 61T^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 5.05T + 83T^{2} \) |
| 89 | \( 1 + 8.12T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−7.86509096196154104532276506310, −7.52454408135749351041509295500, −6.49648213726916228943066043894, −5.78013969503727739470120683348, −4.56750649300663092822050605818, −3.67750189025136652559261958167, −3.22562056391830204977445527184, −2.23023458221086019503308432448, −1.59911187745670014576469031909, 0,
1.59911187745670014576469031909, 2.23023458221086019503308432448, 3.22562056391830204977445527184, 3.67750189025136652559261958167, 4.56750649300663092822050605818, 5.78013969503727739470120683348, 6.49648213726916228943066043894, 7.52454408135749351041509295500, 7.86509096196154104532276506310