L(s) = 1 | + 1.15·2-s + 3-s − 0.655·4-s + 1.84·5-s + 1.15·6-s + 0.698·7-s − 3.07·8-s + 9-s + 2.13·10-s + 0.388·11-s − 0.655·12-s − 13-s + 0.809·14-s + 1.84·15-s − 2.25·16-s + 5.14·17-s + 1.15·18-s + 1.17·19-s − 1.20·20-s + 0.698·21-s + 0.450·22-s + 2.50·23-s − 3.07·24-s − 1.60·25-s − 1.15·26-s + 27-s − 0.458·28-s + ⋯ |
L(s) = 1 | + 0.819·2-s + 0.577·3-s − 0.327·4-s + 0.823·5-s + 0.473·6-s + 0.263·7-s − 1.08·8-s + 0.333·9-s + 0.675·10-s + 0.117·11-s − 0.189·12-s − 0.277·13-s + 0.216·14-s + 0.475·15-s − 0.564·16-s + 1.24·17-s + 0.273·18-s + 0.269·19-s − 0.270·20-s + 0.152·21-s + 0.0960·22-s + 0.522·23-s − 0.628·24-s − 0.321·25-s − 0.227·26-s + 0.192·27-s − 0.0865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.867510799\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.867510799\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 - 0.698T + 7T^{2} \) |
| 11 | \( 1 - 0.388T + 11T^{2} \) |
| 17 | \( 1 - 5.14T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 - 7.84T + 29T^{2} \) |
| 31 | \( 1 - 0.668T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 0.981T + 41T^{2} \) |
| 43 | \( 1 + 7.35T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 6.03T + 53T^{2} \) |
| 59 | \( 1 + 4.69T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 - 7.71T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 17.4T + 79T^{2} \) |
| 83 | \( 1 - 3.30T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.352987962746470796775641553869, −7.893309396253168981758843995920, −6.76438133486295951988175697024, −6.13266624329649731461614124131, −5.21744408862807015949148664230, −4.85158942664065178521567971477, −3.77046367806067099552652412776, −3.11209966007511107756815607353, −2.23059159410367274011863686363, −1.02358543540102969884618772498,
1.02358543540102969884618772498, 2.23059159410367274011863686363, 3.11209966007511107756815607353, 3.77046367806067099552652412776, 4.85158942664065178521567971477, 5.21744408862807015949148664230, 6.13266624329649731461614124131, 6.76438133486295951988175697024, 7.893309396253168981758843995920, 8.352987962746470796775641553869