| L(s) = 1 | + 2-s − 2.56·3-s + 4-s + 1.56·5-s − 2.56·6-s − 7-s + 8-s + 3.56·9-s + 1.56·10-s − 6.56·11-s − 2.56·12-s + 0.438·13-s − 14-s − 4·15-s + 16-s + 2·17-s + 3.56·18-s + 1.56·20-s + 2.56·21-s − 6.56·22-s + 3.56·23-s − 2.56·24-s − 2.56·25-s + 0.438·26-s − 1.43·27-s − 28-s + 10.2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.47·3-s + 0.5·4-s + 0.698·5-s − 1.04·6-s − 0.377·7-s + 0.353·8-s + 1.18·9-s + 0.493·10-s − 1.97·11-s − 0.739·12-s + 0.121·13-s − 0.267·14-s − 1.03·15-s + 0.250·16-s + 0.485·17-s + 0.839·18-s + 0.349·20-s + 0.558·21-s − 1.39·22-s + 0.742·23-s − 0.522·24-s − 0.512·25-s + 0.0859·26-s − 0.276·27-s − 0.188·28-s + 1.90·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 + 6.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 3.56T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 7.68T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 + 9.12T + 53T^{2} \) |
| 59 | \( 1 + 11T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 3.68T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 - 4.56T + 73T^{2} \) |
| 79 | \( 1 - 2.24T + 79T^{2} \) |
| 83 | \( 1 + T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 3.43T + 97T^{2} \) |
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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
−7.67612270193975430926223395278, −6.83648146032189259053632856543, −6.08746778726754815952129415524, −5.79809960930207723972390539998, −4.86645487900528445764371927722, −4.73939206786607909020429387980, −3.20169500745029864967524845650, −2.56603192282416082164874650039, −1.28200103587936256617730006073, 0,
1.28200103587936256617730006073, 2.56603192282416082164874650039, 3.20169500745029864967524845650, 4.73939206786607909020429387980, 4.86645487900528445764371927722, 5.79809960930207723972390539998, 6.08746778726754815952129415524, 6.83648146032189259053632856543, 7.67612270193975430926223395278
