L(s) = 1 | + (−1.24 − 0.678i)2-s + 1.61i·3-s + (1.08 + 1.68i)4-s + 5-s + (1.09 − 1.99i)6-s + (2.13 + 1.56i)7-s + (−0.198 − 2.82i)8-s + 0.405·9-s + (−1.24 − 0.678i)10-s − 6.01·11-s + (−2.71 + 1.73i)12-s + 4.25·13-s + (−1.58 − 3.38i)14-s + 1.61i·15-s + (−1.66 + 3.63i)16-s + 5.42i·17-s + ⋯ |
L(s) = 1 | + (−0.877 − 0.479i)2-s + 0.929i·3-s + (0.540 + 0.841i)4-s + 0.447·5-s + (0.445 − 0.816i)6-s + (0.806 + 0.590i)7-s + (−0.0702 − 0.997i)8-s + 0.135·9-s + (−0.392 − 0.214i)10-s − 1.81·11-s + (−0.782 + 0.502i)12-s + 1.17·13-s + (−0.424 − 0.905i)14-s + 0.415i·15-s + (−0.416 + 0.909i)16-s + 1.31i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.848086 + 0.468313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.848086 + 0.468313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.24 + 0.678i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.13 - 1.56i)T \) |
good | 3 | \( 1 - 1.61iT - 3T^{2} \) |
| 11 | \( 1 + 6.01T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 - 5.42iT - 17T^{2} \) |
| 19 | \( 1 + 4.53iT - 19T^{2} \) |
| 23 | \( 1 - 5.19iT - 23T^{2} \) |
| 29 | \( 1 - 0.376iT - 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 + 0.372iT - 37T^{2} \) |
| 41 | \( 1 + 5.75iT - 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 - 3.86T + 47T^{2} \) |
| 53 | \( 1 + 10.2iT - 53T^{2} \) |
| 59 | \( 1 - 10.5iT - 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 + 0.782T + 67T^{2} \) |
| 71 | \( 1 + 15.3iT - 71T^{2} \) |
| 73 | \( 1 + 8.77iT - 73T^{2} \) |
| 79 | \( 1 + 8.74iT - 79T^{2} \) |
| 83 | \( 1 + 8.42iT - 83T^{2} \) |
| 89 | \( 1 + 1.94iT - 89T^{2} \) |
| 97 | \( 1 + 3.14iT - 97T^{2} \) |
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
−11.61876391910033328384390105321, −10.62331760254483779029190155859, −10.42796274666470429339517796156, −9.179336249935187189249612433876, −8.472679622387953838550753990495, −7.52439859745685008899288386372, −5.91720289636413461743980600682, −4.76305802813230474189115975069, −3.32848696423407492408787933545, −1.88084607191702109434365472577,
1.07602007705532920140135810520, 2.42693316664145583197368751989, 4.86307616371084992503225471137, 5.98542634528804732288583988500, 7.03796878053652777314100894616, 7.88364227326854523015111067615, 8.422838795404406090924408491415, 9.927829016606157565567409861930, 10.58174731505343437904212045263, 11.47689785945099171541326384919