L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s − 4·7-s − 8-s + 9-s + 2·10-s + 11-s + 12-s + 4·14-s − 2·15-s + 16-s + 17-s − 18-s + 4·19-s − 2·20-s − 4·21-s − 22-s − 4·23-s − 24-s − 25-s + 27-s − 4·28-s + 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.872·21-s − 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.230883582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230883582\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.97033230591003, −12.68775470429601, −12.03597976264390, −11.70620421625248, −11.40826248280426, −10.43139402176500, −10.26138456360258, −9.731636959628336, −9.301594767724680, −8.955600048264292, −8.317770146234721, −7.789701187855743, −7.551711988092390, −6.969600994624235, −6.419658694517495, −6.038019871734625, −5.439598901958467, −4.472013037632503, −4.096478659445665, −3.458579801429504, −3.069931858231040, −2.609027211528101, −1.811424671628622, −0.9659723863428163, −0.4088914034603153,
0.4088914034603153, 0.9659723863428163, 1.811424671628622, 2.609027211528101, 3.069931858231040, 3.458579801429504, 4.096478659445665, 4.472013037632503, 5.439598901958467, 6.038019871734625, 6.419658694517495, 6.969600994624235, 7.551711988092390, 7.789701187855743, 8.317770146234721, 8.955600048264292, 9.301594767724680, 9.731636959628336, 10.26138456360258, 10.43139402176500, 11.40826248280426, 11.70620421625248, 12.03597976264390, 12.68775470429601, 12.97033230591003