| L(s) = 1 | − 0.746·3-s + 7-s − 2.44·9-s − 5.90·11-s − 3.20·13-s + 2.14·17-s + 3.56·19-s − 0.746·21-s − 3.75·23-s + 4.06·27-s − 6.61·29-s + 5.79·31-s + 4.41·33-s − 0.623·37-s + 2.39·39-s − 5.43·41-s − 12.6·43-s + 4.31·47-s + 49-s − 1.60·51-s − 2.11·53-s − 2.66·57-s + 7.01·59-s + 1.86·61-s − 2.44·63-s + 6.88·67-s + 2.80·69-s + ⋯ |
| L(s) = 1 | − 0.431·3-s + 0.377·7-s − 0.814·9-s − 1.78·11-s − 0.890·13-s + 0.521·17-s + 0.818·19-s − 0.163·21-s − 0.782·23-s + 0.782·27-s − 1.22·29-s + 1.04·31-s + 0.767·33-s − 0.102·37-s + 0.383·39-s − 0.848·41-s − 1.93·43-s + 0.629·47-s + 0.142·49-s − 0.224·51-s − 0.290·53-s − 0.352·57-s + 0.913·59-s + 0.238·61-s − 0.307·63-s + 0.841·67-s + 0.337·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8599336367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8599336367\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 3 | \( 1 + 0.746T + 3T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 - 2.14T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 0.623T + 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 4.31T + 47T^{2} \) |
| 53 | \( 1 + 2.11T + 53T^{2} \) |
| 59 | \( 1 - 7.01T + 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 - 1.81T + 71T^{2} \) |
| 73 | \( 1 + 6.11T + 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.65T + 97T^{2} \) |
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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
−7.992853900984805095641685937180, −7.62377117401326280960012460858, −6.72351458073164828425916685068, −5.77416908071946167371869693667, −5.20976745522362848144766147250, −4.87192370300687256002653892092, −3.55914579730352392173297217791, −2.77070456896613347383902764992, −1.98670173164360505956783706963, −0.47842812060861252502474953031,
0.47842812060861252502474953031, 1.98670173164360505956783706963, 2.77070456896613347383902764992, 3.55914579730352392173297217791, 4.87192370300687256002653892092, 5.20976745522362848144766147250, 5.77416908071946167371869693667, 6.72351458073164828425916685068, 7.62377117401326280960012460858, 7.992853900984805095641685937180
