| L(s) = 1 | − 5-s + 7-s + 3.12·11-s + 2·13-s + 3.12·17-s − 1.12·19-s − 3.12·23-s + 25-s − 5.12·29-s + 6.24·31-s − 35-s + 7.12·37-s − 8.24·41-s − 1.12·43-s − 1.12·47-s + 49-s + 1.12·53-s − 3.12·55-s + 4·59-s + 11.1·61-s − 2·65-s − 1.12·67-s + 6·71-s − 4.24·73-s + 3.12·77-s + 8·79-s − 3.12·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.941·11-s + 0.554·13-s + 0.757·17-s − 0.257·19-s − 0.651·23-s + 0.200·25-s − 0.951·29-s + 1.12·31-s − 0.169·35-s + 1.17·37-s − 1.28·41-s − 0.171·43-s − 0.163·47-s + 0.142·49-s + 0.154·53-s − 0.421·55-s + 0.520·59-s + 1.42·61-s − 0.248·65-s − 0.137·67-s + 0.712·71-s − 0.496·73-s + 0.355·77-s + 0.900·79-s − 0.338·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.894078745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.894078745\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 5.12T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 1.12T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.12T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 4.24T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.764729744065793123026603377491, −8.196876056835412993373062465216, −7.46812669097100297731122038735, −6.57397941473870578427004275844, −5.89365713705756094355690727305, −4.89798992956906952381733717286, −4.01962470029307670187176637030, −3.37091904781309077339467492228, −2.03795540012485718792439014116, −0.912033886118454680554444745457,
0.912033886118454680554444745457, 2.03795540012485718792439014116, 3.37091904781309077339467492228, 4.01962470029307670187176637030, 4.89798992956906952381733717286, 5.89365713705756094355690727305, 6.57397941473870578427004275844, 7.46812669097100297731122038735, 8.196876056835412993373062465216, 8.764729744065793123026603377491
