| L(s) = 1 | − 2.36·3-s + 1.80·5-s − 2.68·7-s + 2.57·9-s − 3.21·11-s − 3.17·13-s − 4.25·15-s − 4.79·17-s − 1.49·19-s + 6.34·21-s − 1.06·23-s − 1.75·25-s + 1.01·27-s − 6.65·29-s − 10.4·31-s + 7.59·33-s − 4.84·35-s + 6.90·37-s + 7.49·39-s − 7.98·43-s + 4.63·45-s + 4.19·47-s + 0.226·49-s + 11.3·51-s − 2.91·53-s − 5.80·55-s + 3.53·57-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.806·5-s − 1.01·7-s + 0.856·9-s − 0.970·11-s − 0.880·13-s − 1.09·15-s − 1.16·17-s − 0.343·19-s + 1.38·21-s − 0.222·23-s − 0.350·25-s + 0.195·27-s − 1.23·29-s − 1.87·31-s + 1.32·33-s − 0.819·35-s + 1.13·37-s + 1.19·39-s − 1.21·43-s + 0.690·45-s + 0.612·47-s + 0.0324·49-s + 1.58·51-s − 0.401·53-s − 0.782·55-s + 0.468·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1695317438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1695317438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 2.68T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 + 3.17T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 6.90T + 37T^{2} \) |
| 43 | \( 1 + 7.98T + 43T^{2} \) |
| 47 | \( 1 - 4.19T + 47T^{2} \) |
| 53 | \( 1 + 2.91T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 9.15T + 67T^{2} \) |
| 71 | \( 1 - 5.22T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 - 9.29T + 79T^{2} \) |
| 83 | \( 1 + 0.226T + 83T^{2} \) |
| 89 | \( 1 - 4.07T + 89T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−7.72326096891175756737210579067, −7.11145926491702663076828387551, −6.34061324777720643895012720464, −5.93626036321147975122449805557, −5.27876408234932751616615171347, −4.68730094302645135086779050411, −3.65651894689789976609279476668, −2.56652868363434387956820837294, −1.85201501040115832272182463342, −0.21141238111107290605333225610,
0.21141238111107290605333225610, 1.85201501040115832272182463342, 2.56652868363434387956820837294, 3.65651894689789976609279476668, 4.68730094302645135086779050411, 5.27876408234932751616615171347, 5.93626036321147975122449805557, 6.34061324777720643895012720464, 7.11145926491702663076828387551, 7.72326096891175756737210579067
