| L(s) = 1 | + 1.20·3-s + 3.49·5-s + 0.744·7-s − 1.54·9-s − 6.28·11-s − 4.06·13-s + 4.20·15-s − 1.60·17-s − 2.14·19-s + 0.897·21-s − 9.25·23-s + 7.18·25-s − 5.48·27-s + 3.13·29-s + 3.15·31-s − 7.57·33-s + 2.59·35-s + 4.19·37-s − 4.90·39-s − 4.52·41-s − 6.99·43-s − 5.40·45-s + 4.82·47-s − 6.44·49-s − 1.92·51-s + 3.71·53-s − 21.9·55-s + ⋯ |
| L(s) = 1 | + 0.695·3-s + 1.56·5-s + 0.281·7-s − 0.515·9-s − 1.89·11-s − 1.12·13-s + 1.08·15-s − 0.388·17-s − 0.491·19-s + 0.195·21-s − 1.92·23-s + 1.43·25-s − 1.05·27-s + 0.581·29-s + 0.566·31-s − 1.31·33-s + 0.439·35-s + 0.690·37-s − 0.785·39-s − 0.706·41-s − 1.06·43-s − 0.805·45-s + 0.703·47-s − 0.920·49-s − 0.270·51-s + 0.509·53-s − 2.95·55-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 | \( 1 - 1.20T + 3T^{2} \) |
| 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 - 0.744T + 7T^{2} \) |
| 11 | \( 1 + 6.28T + 11T^{2} \) |
| 13 | \( 1 + 4.06T + 13T^{2} \) |
| 17 | \( 1 + 1.60T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 + 9.25T + 23T^{2} \) |
| 29 | \( 1 - 3.13T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 + 4.52T + 41T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 + 15.7T + 71T^{2} \) |
| 73 | \( 1 + 1.64T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 4.74T + 83T^{2} \) |
| 89 | \( 1 + 7.95T + 89T^{2} \) |
| 97 | \( 1 + 5.49T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.167930524165074988493416244805, −7.61432254045727003681435641400, −6.53791691559402482719881401968, −5.78224014627511457195940281297, −5.22860774637844101476211510343, −4.45003553157400118897283395288, −3.01674771650957396072837151365, −2.37488104136698565565192802108, −1.96143848324252131397511520523, 0,
1.96143848324252131397511520523, 2.37488104136698565565192802108, 3.01674771650957396072837151365, 4.45003553157400118897283395288, 5.22860774637844101476211510343, 5.78224014627511457195940281297, 6.53791691559402482719881401968, 7.61432254045727003681435641400, 8.167930524165074988493416244805
