| L(s) = 1 | + 3-s + 2.46·7-s + 9-s + 4.19·11-s + 3.26·13-s + 7.73·17-s + 0.732·19-s + 2.46·21-s + 23-s + 27-s + 7.19·29-s − 31-s + 4.19·33-s + 11.3·37-s + 3.26·39-s − 7.73·41-s − 3.46·43-s − 0.732·47-s − 0.928·49-s + 7.73·51-s − 6.66·53-s + 0.732·57-s − 7.19·59-s − 10.7·61-s + 2.46·63-s − 5·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.931·7-s + 0.333·9-s + 1.26·11-s + 0.906·13-s + 1.87·17-s + 0.167·19-s + 0.537·21-s + 0.208·23-s + 0.192·27-s + 1.33·29-s − 0.179·31-s + 0.730·33-s + 1.87·37-s + 0.523·39-s − 1.20·41-s − 0.528·43-s − 0.106·47-s − 0.132·49-s + 1.08·51-s − 0.914·53-s + 0.0969·57-s − 0.936·59-s − 1.37·61-s + 0.310·63-s − 0.610·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.863175870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.863175870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 - 4.19T + 11T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 - 7.73T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 29 | \( 1 - 7.19T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 0.732T + 47T^{2} \) |
| 53 | \( 1 + 6.66T + 53T^{2} \) |
| 59 | \( 1 + 7.19T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.064469453634941485872001694336, −7.44822699813276739718488383797, −6.51891153423254961118135236775, −5.96176092558733571222381817052, −4.99894245912163190656978098211, −4.32932100485992812867730832759, −3.50817757396754464785759795382, −2.89562950123847344779858423142, −1.49679278795648747522742976324, −1.20493256948798574455525916220,
1.20493256948798574455525916220, 1.49679278795648747522742976324, 2.89562950123847344779858423142, 3.50817757396754464785759795382, 4.32932100485992812867730832759, 4.99894245912163190656978098211, 5.96176092558733571222381817052, 6.51891153423254961118135236775, 7.44822699813276739718488383797, 8.064469453634941485872001694336
