L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 3·11-s + 12-s + 5·13-s + 14-s + 16-s − 18-s − 4·19-s − 21-s − 3·22-s + 6·23-s − 24-s − 5·26-s + 27-s − 28-s − 6·29-s + 31-s − 32-s + 3·33-s + 36-s + 5·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.218·21-s − 0.639·22-s + 1.25·23-s − 0.204·24-s − 0.980·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.179·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + 0.821·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.920620957\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.920620957\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.538415620155690647929044459957, −7.69054301412611961380311191974, −6.91836963607233262066916811908, −6.34921996263954448770189945788, −5.61906753431831267296914719072, −4.30934907941012244076029563921, −3.67752738769173853980963965360, −2.83150751457724296351699408798, −1.78643073962409431486138684180, −0.872338448708764522787732027364,
0.872338448708764522787732027364, 1.78643073962409431486138684180, 2.83150751457724296351699408798, 3.67752738769173853980963965360, 4.30934907941012244076029563921, 5.61906753431831267296914719072, 6.34921996263954448770189945788, 6.91836963607233262066916811908, 7.69054301412611961380311191974, 8.538415620155690647929044459957