| L(s) = 1 | − 1.22·3-s − 3.14·5-s − 0.136·7-s − 1.50·9-s + 0.905·11-s − 0.123·13-s + 3.84·15-s + 1.26·17-s + 2.13·19-s + 0.166·21-s − 6.64·23-s + 4.91·25-s + 5.50·27-s + 5.36·29-s + 9.78·31-s − 1.10·33-s + 0.428·35-s + 5.76·37-s + 0.151·39-s + 6.43·41-s + 3.18·43-s + 4.74·45-s − 12.9·47-s − 6.98·49-s − 1.54·51-s + 3.90·53-s − 2.85·55-s + ⋯ |
| L(s) = 1 | − 0.705·3-s − 1.40·5-s − 0.0514·7-s − 0.502·9-s + 0.272·11-s − 0.0343·13-s + 0.993·15-s + 0.305·17-s + 0.489·19-s + 0.0363·21-s − 1.38·23-s + 0.982·25-s + 1.05·27-s + 0.996·29-s + 1.75·31-s − 0.192·33-s + 0.0724·35-s + 0.948·37-s + 0.0242·39-s + 1.00·41-s + 0.486·43-s + 0.707·45-s − 1.89·47-s − 0.997·49-s − 0.215·51-s + 0.536·53-s − 0.384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 7 | \( 1 + 0.136T + 7T^{2} \) |
| 11 | \( 1 - 0.905T + 11T^{2} \) |
| 13 | \( 1 + 0.123T + 13T^{2} \) |
| 17 | \( 1 - 1.26T + 17T^{2} \) |
| 19 | \( 1 - 2.13T + 19T^{2} \) |
| 23 | \( 1 + 6.64T + 23T^{2} \) |
| 29 | \( 1 - 5.36T + 29T^{2} \) |
| 31 | \( 1 - 9.78T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 - 6.43T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 + 8.15T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 4.89T + 67T^{2} \) |
| 71 | \( 1 + 4.32T + 71T^{2} \) |
| 73 | \( 1 - 5.64T + 73T^{2} \) |
| 79 | \( 1 + 1.43T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.980580832286562607342417775950, −7.60010553851710736551611927410, −6.40030934081084772645521791278, −6.13040445667835866718489147518, −4.92785167242043266800778900598, −4.41515758821415726869802490128, −3.48849074077679993475903758379, −2.69177035856905714814283333913, −1.05713099778181009484973797491, 0,
1.05713099778181009484973797491, 2.69177035856905714814283333913, 3.48849074077679993475903758379, 4.41515758821415726869802490128, 4.92785167242043266800778900598, 6.13040445667835866718489147518, 6.40030934081084772645521791278, 7.60010553851710736551611927410, 7.980580832286562607342417775950
