L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 1.32·7-s − 8-s + 9-s − 10-s + 4.61·11-s + 12-s − 1.32·14-s + 15-s + 16-s + 4·17-s − 18-s − 2.29·19-s + 20-s + 1.32·21-s − 4.61·22-s + 8.66·23-s − 24-s + 25-s + 27-s + 1.32·28-s − 2.02·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.499·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.39·11-s + 0.288·12-s − 0.353·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 0.525·19-s + 0.223·20-s + 0.288·21-s − 0.983·22-s + 1.80·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.249·28-s − 0.375·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.543248150\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.543248150\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 - 8.66T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 6.80T + 37T^{2} \) |
| 41 | \( 1 + 4.64T + 41T^{2} \) |
| 43 | \( 1 - 8.60T + 43T^{2} \) |
| 47 | \( 1 + 9.10T + 47T^{2} \) |
| 53 | \( 1 - 0.826T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 - 0.535T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.28T + 73T^{2} \) |
| 79 | \( 1 - 2.96T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 8.81T + 97T^{2} \) |
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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
−8.491753737729213031857048593391, −7.60366142317056562893763378960, −6.85853999466127453246747289506, −6.36095960957383580750210860055, −5.34087396705328820126080238679, −4.51887431100614028282246193595, −3.52103616228626148594094929593, −2.76131231924620139097082609279, −1.66212441934029509137155483068, −1.05653749710363503837799324999,
1.05653749710363503837799324999, 1.66212441934029509137155483068, 2.76131231924620139097082609279, 3.52103616228626148594094929593, 4.51887431100614028282246193595, 5.34087396705328820126080238679, 6.36095960957383580750210860055, 6.85853999466127453246747289506, 7.60366142317056562893763378960, 8.491753737729213031857048593391