L(s) = 1 | − 1.18·2-s − 3-s − 0.596·4-s − 2.40·5-s + 1.18·6-s − 0.655·7-s + 3.07·8-s + 9-s + 2.84·10-s − 6.56·11-s + 0.596·12-s − 4.80·13-s + 0.776·14-s + 2.40·15-s − 2.45·16-s + 17-s − 1.18·18-s + 5.62·19-s + 1.43·20-s + 0.655·21-s + 7.77·22-s + 4.12·23-s − 3.07·24-s + 0.766·25-s + 5.69·26-s − 27-s + 0.390·28-s + ⋯ |
L(s) = 1 | − 0.837·2-s − 0.577·3-s − 0.298·4-s − 1.07·5-s + 0.483·6-s − 0.247·7-s + 1.08·8-s + 0.333·9-s + 0.899·10-s − 1.97·11-s + 0.172·12-s − 1.33·13-s + 0.207·14-s + 0.620·15-s − 0.612·16-s + 0.242·17-s − 0.279·18-s + 1.29·19-s + 0.320·20-s + 0.143·21-s + 1.65·22-s + 0.860·23-s − 0.627·24-s + 0.153·25-s + 1.11·26-s − 0.192·27-s + 0.0738·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 7 | \( 1 + 0.655T + 7T^{2} \) |
| 11 | \( 1 + 6.56T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 19 | \( 1 - 5.62T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 - 1.32T + 41T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 - 7.91T + 59T^{2} \) |
| 61 | \( 1 - 3.43T + 61T^{2} \) |
| 67 | \( 1 - 3.90T + 67T^{2} \) |
| 71 | \( 1 - 7.52T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 83 | \( 1 + 8.23T + 83T^{2} \) |
| 89 | \( 1 - 18.2T + 89T^{2} \) |
| 97 | \( 1 - 2.31T + 97T^{2} \) |
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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
−7.897079586920060969422102104588, −7.49370943370177470729500225308, −7.13072546063741024424971501226, −5.64180668641334751246622747780, −5.04674062493778259415645690956, −4.49232531996032450840573296327, −3.36072465894549933298300731928, −2.42172576151663452497379101782, −0.816277219083521955974208711480, 0,
0.816277219083521955974208711480, 2.42172576151663452497379101782, 3.36072465894549933298300731928, 4.49232531996032450840573296327, 5.04674062493778259415645690956, 5.64180668641334751246622747780, 7.13072546063741024424971501226, 7.49370943370177470729500225308, 7.897079586920060969422102104588