| L(s) = 1 | − 2-s − 2.56·3-s + 4-s + 2.56·6-s + 1.56·7-s − 8-s + 3.56·9-s − 11-s − 2.56·12-s − 3.56·13-s − 1.56·14-s + 16-s − 1.43·17-s − 3.56·18-s + 3·19-s − 4·21-s + 22-s − 23-s + 2.56·24-s + 3.56·26-s − 1.43·27-s + 1.56·28-s + 5.56·29-s + 3.12·31-s − 32-s + 2.56·33-s + 1.43·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.47·3-s + 0.5·4-s + 1.04·6-s + 0.590·7-s − 0.353·8-s + 1.18·9-s − 0.301·11-s − 0.739·12-s − 0.987·13-s − 0.417·14-s + 0.250·16-s − 0.348·17-s − 0.839·18-s + 0.688·19-s − 0.872·21-s + 0.213·22-s − 0.208·23-s + 0.522·24-s + 0.698·26-s − 0.276·27-s + 0.295·28-s + 1.03·29-s + 0.560·31-s − 0.176·32-s + 0.445·33-s + 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 4.68T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 0.876T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 1.43T + 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 2.12T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 - 2.12T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 8.24T + 97T^{2} \) |
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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
−9.672952104854083319637490301114, −8.513963836835605424578674329667, −7.68112746023625407789495020462, −6.87453993432094638869469723917, −6.08470739443190062512671795373, −5.15197346640548577991566128149, −4.53423494173869407757499346348, −2.79339918431607073807434057803, −1.36787880357210621874848810705, 0,
1.36787880357210621874848810705, 2.79339918431607073807434057803, 4.53423494173869407757499346348, 5.15197346640548577991566128149, 6.08470739443190062512671795373, 6.87453993432094638869469723917, 7.68112746023625407789495020462, 8.513963836835605424578674329667, 9.672952104854083319637490301114
