| L(s) = 1 | − 5-s − 7-s − 3·11-s + 5·17-s − 2·19-s − 2·23-s + 25-s + 6·29-s + 35-s + 6·37-s − 4·41-s − 43-s + 2·47-s − 6·49-s + 7·53-s + 3·55-s + 15·61-s + 5·67-s − 3·71-s + 4·73-s + 3·77-s + 2·79-s + 6·83-s − 5·85-s + 10·89-s + 2·95-s − 4·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.904·11-s + 1.21·17-s − 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s + 0.169·35-s + 0.986·37-s − 0.624·41-s − 0.152·43-s + 0.291·47-s − 6/7·49-s + 0.961·53-s + 0.404·55-s + 1.92·61-s + 0.610·67-s − 0.356·71-s + 0.468·73-s + 0.341·77-s + 0.225·79-s + 0.658·83-s − 0.542·85-s + 1.05·89-s + 0.205·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.654755059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.654755059\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.97521749219526, −13.51249485071618, −12.95717044089206, −12.58869114115964, −11.99590992234021, −11.68606599865016, −10.95539487305252, −10.51342511723832, −9.957019476622500, −9.707313896562927, −8.898122932638655, −8.253949014701964, −8.030443655034683, −7.456140882999812, −6.782488123281453, −6.360711765974628, −5.611444255623152, −5.207284441016755, −4.556908544277416, −3.891137066079249, −3.360105965273828, −2.715318934311198, −2.171073734024526, −1.148709438884631, −0.4632835141785357,
0.4632835141785357, 1.148709438884631, 2.171073734024526, 2.715318934311198, 3.360105965273828, 3.891137066079249, 4.556908544277416, 5.207284441016755, 5.611444255623152, 6.360711765974628, 6.782488123281453, 7.456140882999812, 8.030443655034683, 8.253949014701964, 8.898122932638655, 9.707313896562927, 9.957019476622500, 10.51342511723832, 10.95539487305252, 11.68606599865016, 11.99590992234021, 12.58869114115964, 12.95717044089206, 13.51249485071618, 13.97521749219526
