| L(s) = 1 | + 2.60·3-s − 1.33·7-s + 3.80·9-s − 1.09·11-s − 3.17·13-s − 17-s − 2.75·19-s − 3.47·21-s + 3.57·23-s + 2.08·27-s − 0.180·29-s − 0.816·31-s − 2.85·33-s − 8.44·37-s − 8.26·39-s − 7.97·41-s + 6.54·43-s + 0.576·47-s − 5.22·49-s − 2.60·51-s − 7.84·53-s − 7.18·57-s + 9.76·59-s − 5.21·61-s − 5.06·63-s − 2.64·67-s + 9.31·69-s + ⋯ |
| L(s) = 1 | + 1.50·3-s − 0.503·7-s + 1.26·9-s − 0.330·11-s − 0.879·13-s − 0.242·17-s − 0.632·19-s − 0.758·21-s + 0.744·23-s + 0.402·27-s − 0.0334·29-s − 0.146·31-s − 0.497·33-s − 1.38·37-s − 1.32·39-s − 1.24·41-s + 0.998·43-s + 0.0840·47-s − 0.746·49-s − 0.365·51-s − 1.07·53-s − 0.952·57-s + 1.27·59-s − 0.667·61-s − 0.638·63-s − 0.323·67-s + 1.12·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 7 | \( 1 + 1.33T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 19 | \( 1 + 2.75T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 + 0.180T + 29T^{2} \) |
| 31 | \( 1 + 0.816T + 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 + 7.97T + 41T^{2} \) |
| 43 | \( 1 - 6.54T + 43T^{2} \) |
| 47 | \( 1 - 0.576T + 47T^{2} \) |
| 53 | \( 1 + 7.84T + 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 + 5.21T + 61T^{2} \) |
| 67 | \( 1 + 2.64T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 + 4.66T + 83T^{2} \) |
| 89 | \( 1 - 3.00T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.69501795701655297923587361321, −7.08838438410019358750509007422, −6.46129472935033133091245223732, −5.37642898087720672973205651245, −4.63174468535616946480333981026, −3.75973845814794589811086043696, −3.07495029815361337046772410980, −2.45848726218364177877998395406, −1.63813823551447971298577533024, 0,
1.63813823551447971298577533024, 2.45848726218364177877998395406, 3.07495029815361337046772410980, 3.75973845814794589811086043696, 4.63174468535616946480333981026, 5.37642898087720672973205651245, 6.46129472935033133091245223732, 7.08838438410019358750509007422, 7.69501795701655297923587361321
