| L(s) = 1 | + 2-s + 3·3-s + 4-s − 3·5-s + 3·6-s − 2·7-s + 8-s + 6·9-s − 3·10-s + 11-s + 3·12-s + 3·13-s − 2·14-s − 9·15-s + 16-s + 4·17-s + 6·18-s + 8·19-s − 3·20-s − 6·21-s + 22-s + 3·24-s + 4·25-s + 3·26-s + 9·27-s − 2·28-s − 9·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.34·5-s + 1.22·6-s − 0.755·7-s + 0.353·8-s + 2·9-s − 0.948·10-s + 0.301·11-s + 0.866·12-s + 0.832·13-s − 0.534·14-s − 2.32·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s + 1.83·19-s − 0.670·20-s − 1.30·21-s + 0.213·22-s + 0.612·24-s + 4/5·25-s + 0.588·26-s + 1.73·27-s − 0.377·28-s − 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.059739595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.059739595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−9.386775863413184661004558855327, −8.312085404745405461717701868441, −7.80131581534615262996684417458, −7.23778336587483545649646561951, −6.25958344189210852249588590955, −4.95604830522157167594284576961, −3.81404797070161537710517021094, −3.47961283259009069356933090452, −2.88939356621529724855392570973, −1.32364216049448157109471695089,
1.32364216049448157109471695089, 2.88939356621529724855392570973, 3.47961283259009069356933090452, 3.81404797070161537710517021094, 4.95604830522157167594284576961, 6.25958344189210852249588590955, 7.23778336587483545649646561951, 7.80131581534615262996684417458, 8.312085404745405461717701868441, 9.386775863413184661004558855327
