| L(s) = 1 | + 1.12·3-s − 1.74·9-s − 4.94·11-s − 4.64·13-s + 5.72·17-s + 7.53·19-s + 6.87·23-s − 5.31·27-s + 8.11·29-s − 3.87·31-s − 5.54·33-s − 5.18·37-s − 5.20·39-s − 9.45·41-s + 0.706·43-s − 2.20·47-s + 6.42·51-s + 1.32·53-s + 8.45·57-s − 3.94·59-s − 6.07·61-s + 8.24·67-s + 7.71·69-s − 4.50·71-s − 2.93·73-s − 11.0·79-s − 0.746·81-s + ⋯ |
| L(s) = 1 | + 0.647·3-s − 0.580·9-s − 1.49·11-s − 1.28·13-s + 1.38·17-s + 1.72·19-s + 1.43·23-s − 1.02·27-s + 1.50·29-s − 0.696·31-s − 0.965·33-s − 0.852·37-s − 0.833·39-s − 1.47·41-s + 0.107·43-s − 0.321·47-s + 0.899·51-s + 0.181·53-s + 1.12·57-s − 0.513·59-s − 0.777·61-s + 1.00·67-s + 0.928·69-s − 0.535·71-s − 0.343·73-s − 1.24·79-s − 0.0829·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.12T + 3T^{2} \) |
| 11 | \( 1 + 4.94T + 11T^{2} \) |
| 13 | \( 1 + 4.64T + 13T^{2} \) |
| 17 | \( 1 - 5.72T + 17T^{2} \) |
| 19 | \( 1 - 7.53T + 19T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 - 8.11T + 29T^{2} \) |
| 31 | \( 1 + 3.87T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + 9.45T + 41T^{2} \) |
| 43 | \( 1 - 0.706T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 - 8.24T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 - 5.27T + 83T^{2} \) |
| 89 | \( 1 + 9.49T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−7.47636644869085966420732047693, −6.93204402514400832140177431670, −5.72428784714890826858633985695, −5.17564234588725175256447396248, −4.87054026574766547952996105588, −3.38449856969761141091262839115, −3.05007284855788291218180407007, −2.45865475959891193059177389260, −1.25234827823432171687725604638, 0,
1.25234827823432171687725604638, 2.45865475959891193059177389260, 3.05007284855788291218180407007, 3.38449856969761141091262839115, 4.87054026574766547952996105588, 5.17564234588725175256447396248, 5.72428784714890826858633985695, 6.93204402514400832140177431670, 7.47636644869085966420732047693
