| L(s) = 1 | − 2.41·3-s + 2.41·7-s + 2.82·9-s − 1.41·11-s + 1.82·13-s + 17-s + 19-s − 5.82·21-s − 5.24·23-s + 0.414·27-s + 3.82·29-s − 3.41·31-s + 3.41·33-s + 5.17·37-s − 4.41·39-s − 7.07·41-s − 2.24·43-s − 8·47-s − 1.17·49-s − 2.41·51-s − 1.82·53-s − 2.41·57-s − 14.4·59-s + 3.41·61-s + 6.82·63-s + 6.07·67-s + 12.6·69-s + ⋯ |
| L(s) = 1 | − 1.39·3-s + 0.912·7-s + 0.942·9-s − 0.426·11-s + 0.507·13-s + 0.242·17-s + 0.229·19-s − 1.27·21-s − 1.09·23-s + 0.0797·27-s + 0.710·29-s − 0.613·31-s + 0.594·33-s + 0.850·37-s − 0.706·39-s − 1.10·41-s − 0.341·43-s − 1.16·47-s − 0.167·49-s − 0.338·51-s − 0.251·53-s − 0.319·57-s − 1.87·59-s + 0.437·61-s + 0.860·63-s + 0.741·67-s + 1.52·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 1.82T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 + 5.24T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 + 3.41T + 31T^{2} \) |
| 37 | \( 1 - 5.17T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 1.82T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 3.41T + 61T^{2} \) |
| 67 | \( 1 - 6.07T + 67T^{2} \) |
| 71 | \( 1 + 3.07T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 0.828T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 3.75T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−7.61771084735817423262311289446, −6.51267623496176993291326644980, −6.22547758500510876648110098454, −5.26710488766648331407377673434, −4.99881897894251650911263695977, −4.16711607889211427801941163521, −3.20673599428109798944918002263, −1.98525114463207343413458412028, −1.13896275890434627643294934888, 0,
1.13896275890434627643294934888, 1.98525114463207343413458412028, 3.20673599428109798944918002263, 4.16711607889211427801941163521, 4.99881897894251650911263695977, 5.26710488766648331407377673434, 6.22547758500510876648110098454, 6.51267623496176993291326644980, 7.61771084735817423262311289446
