L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 17.7·5-s − 6·6-s − 13.9·7-s − 8·8-s + 9·9-s + 35.4·10-s − 46.9·11-s + 12·12-s + 12.6·13-s + 27.9·14-s − 53.2·15-s + 16·16-s − 12.7·17-s − 18·18-s + 62.9·19-s − 70.9·20-s − 41.8·21-s + 93.8·22-s + 137.·23-s − 24·24-s + 189.·25-s − 25.2·26-s + 27·27-s − 55.8·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.58·5-s − 0.408·6-s − 0.753·7-s − 0.353·8-s + 0.333·9-s + 1.12·10-s − 1.28·11-s + 0.288·12-s + 0.269·13-s + 0.533·14-s − 0.916·15-s + 0.250·16-s − 0.182·17-s − 0.235·18-s + 0.760·19-s − 0.793·20-s − 0.435·21-s + 0.909·22-s + 1.24·23-s − 0.204·24-s + 1.51·25-s − 0.190·26-s + 0.192·27-s − 0.376·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8443823174\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8443823174\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
good | 5 | \( 1 + 17.7T + 125T^{2} \) |
| 7 | \( 1 + 13.9T + 343T^{2} \) |
| 11 | \( 1 + 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 137.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 36.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 7.66T + 5.06e4T^{2} \) |
| 41 | \( 1 + 108.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 228.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 474.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 645.T + 1.48e5T^{2} \) |
| 61 | \( 1 - 134.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 697.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 276.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 951.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 187.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 702.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 98.6T + 9.12e5T^{2} \) |
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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
−10.95133774131017222614833843198, −10.08584089315568978837063328267, −9.034405271469973938123161763518, −8.156727618349168090290153406943, −7.55867296407187048734915071422, −6.66015679292481497197091492524, −4.98191374080517004570406998287, −3.57622724923529302200239175190, −2.76306141954749802039330896883, −0.64475553699824018271861185793,
0.64475553699824018271861185793, 2.76306141954749802039330896883, 3.57622724923529302200239175190, 4.98191374080517004570406998287, 6.66015679292481497197091492524, 7.55867296407187048734915071422, 8.156727618349168090290153406943, 9.034405271469973938123161763518, 10.08584089315568978837063328267, 10.95133774131017222614833843198