| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 3.23·7-s − 8-s + 9-s + 10-s − 12-s − 3.38·13-s − 3.23·14-s + 15-s + 16-s + 6.09·17-s − 18-s − 1.23·19-s − 20-s − 3.23·21-s + 6.85·23-s + 24-s + 25-s + 3.38·26-s − 27-s + 3.23·28-s + 2.61·29-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 + 1.22·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s − 0.937·13-s − 0.864·14-s + 0.258·15-s + 0.250·16-s + 1.47·17-s − 0.235·18-s − 0.283·19-s − 0.223·20-s − 0.706·21-s + 1.42·23-s + 0.204·24-s + 0.200·25-s + 0.663·26-s − 0.192·27-s + 0.611·28-s + 0.486·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.143547892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143547892\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3.23T + 7T^{2} \) |
| 13 | \( 1 + 3.38T + 13T^{2} \) |
| 17 | \( 1 - 6.09T + 17T^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 - 6.85T + 23T^{2} \) |
| 29 | \( 1 - 2.61T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 + 6.38T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 - 0.0901T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−8.482112172739504241655941987941, −7.74941461330178274104848994311, −7.28430555222644324657491798790, −6.52595582974709996887642585523, −5.24183036563599671736682081842, −5.11240328260593920082804794093, −3.93728582106357798979979787420, −2.85069428754552180810750142096, −1.69554895581313174170166439735, −0.75051979530605171493580269027,
0.75051979530605171493580269027, 1.69554895581313174170166439735, 2.85069428754552180810750142096, 3.93728582106357798979979787420, 5.11240328260593920082804794093, 5.24183036563599671736682081842, 6.52595582974709996887642585523, 7.28430555222644324657491798790, 7.74941461330178274104848994311, 8.482112172739504241655941987941
