| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3.60·7-s + 8-s + 9-s + 10-s + 2.42·11-s + 12-s + 4.98·13-s + 3.60·14-s + 15-s + 16-s + 18-s + 0.319·19-s + 20-s + 3.60·21-s + 2.42·22-s + 5.84·23-s + 24-s + 25-s + 4.98·26-s + 27-s + 3.60·28-s − 3.05·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.36·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.731·11-s + 0.288·12-s + 1.38·13-s + 0.962·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s + 0.0732·19-s + 0.223·20-s + 0.786·21-s + 0.517·22-s + 1.21·23-s + 0.204·24-s + 0.200·25-s + 0.978·26-s + 0.192·27-s + 0.680·28-s − 0.567·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.571520266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.571520266\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 - 4.98T + 13T^{2} \) |
| 19 | \( 1 - 0.319T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 - 6.66T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 8.01T + 53T^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 5.90T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 2.96T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−7.65870846090970924316731656404, −7.10527622032092878002801525131, −6.32846872673464628054699143095, −5.57550802096565093826356694128, −4.99875283880221913043081373615, −4.15751005909440109763583260459, −3.60754737060445536718007492667, −2.72398801610177522381119696432, −1.64046423482509619377263121472, −1.34179197661326182270228486756,
1.34179197661326182270228486756, 1.64046423482509619377263121472, 2.72398801610177522381119696432, 3.60754737060445536718007492667, 4.15751005909440109763583260459, 4.99875283880221913043081373615, 5.57550802096565093826356694128, 6.32846872673464628054699143095, 7.10527622032092878002801525131, 7.65870846090970924316731656404
