| L(s) = 1 | − 4·11-s − 2·13-s − 6·17-s + 4·19-s + 23-s − 2·29-s − 2·37-s − 10·41-s + 4·43-s − 7·49-s − 6·53-s + 4·59-s + 10·61-s + 12·67-s − 8·71-s − 10·73-s + 8·79-s − 4·83-s − 18·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.208·23-s − 0.371·29-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 1.46·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s − 0.439·83-s − 1.90·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.95931464462819, −12.76246807050348, −12.17086500566622, −11.54324311723776, −11.29043710633501, −10.85704637547706, −10.22930360319834, −9.917958684793024, −9.516821760470729, −8.741992115499928, −8.642235342566732, −7.877501120369636, −7.601271786185495, −6.978026900875380, −6.677712405201763, −6.052643501082044, −5.371401829006638, −5.070444906934118, −4.710204413235839, −3.941892766988918, −3.506380133171778, −2.666190283811162, −2.550140393029499, −1.775623581903788, −1.143968169334409, 0, 0,
1.143968169334409, 1.775623581903788, 2.550140393029499, 2.666190283811162, 3.506380133171778, 3.941892766988918, 4.710204413235839, 5.070444906934118, 5.371401829006638, 6.052643501082044, 6.677712405201763, 6.978026900875380, 7.601271786185495, 7.877501120369636, 8.642235342566732, 8.741992115499928, 9.516821760470729, 9.917958684793024, 10.22930360319834, 10.85704637547706, 11.29043710633501, 11.54324311723776, 12.17086500566622, 12.76246807050348, 12.95931464462819
