| L(s) = 1 | + (6.70 − 20.6i)3-s + (67.3 − 48.9i)5-s + (−74.4 − 229. i)7-s + (−184. − 134. i)9-s + (−79.9 + 393. i)11-s + (651. + 473. i)13-s + (−558. − 1.71e3i)15-s + (1.02e3 − 745. i)17-s + (−564. + 1.73e3i)19-s − 5.22e3·21-s − 3.37e3·23-s + (1.17e3 − 3.62e3i)25-s + (257. − 186. i)27-s + (−302. − 931. i)29-s + (−1.70e3 − 1.24e3i)31-s + ⋯ |
| L(s) = 1 | + (0.430 − 1.32i)3-s + (1.20 − 0.875i)5-s + (−0.574 − 1.76i)7-s + (−0.760 − 0.552i)9-s + (−0.199 + 0.979i)11-s + (1.06 + 0.776i)13-s + (−0.641 − 1.97i)15-s + (0.861 − 0.626i)17-s + (−0.358 + 1.10i)19-s − 2.58·21-s − 1.32·23-s + (0.377 − 1.16i)25-s + (0.0679 − 0.0493i)27-s + (−0.0668 − 0.205i)29-s + (−0.319 − 0.232i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.882688 - 2.19193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.882688 - 2.19193i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + (79.9 - 393. i)T \) |
| good | 3 | \( 1 + (-6.70 + 20.6i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + (-67.3 + 48.9i)T + (965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (74.4 + 229. i)T + (-1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-651. - 473. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.02e3 + 745. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (564. - 1.73e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 3.37e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (302. + 931. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.70e3 + 1.24e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-371. - 1.14e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (1.86e3 - 5.74e3i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.97e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.93e3 + 1.82e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.36e4 - 9.89e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.64e3 - 5.07e3i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-7.13e3 + 5.18e3i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 2.59e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-2.57e4 + 1.87e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.36e3 - 4.20e3i)T + (-1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-5.23e4 - 3.80e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (2.83e4 - 2.05e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + 5.00e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (2.72e4 + 1.98e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
| 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
−13.02840311525799671432275746978, −12.20529137838960248473174727625, −10.33657038470424756916539240453, −9.514669932988743253116735438471, −8.036141011034969045608498542197, −7.05525062866968332499176621323, −5.99176857028319611420413871735, −4.08206173985093145179157034651, −1.89554073192896241350071476252, −0.978050462461175040718705402205,
2.52066265795792516445212312656, 3.42849610833683171284323889566, 5.61350910048538822985528472716, 6.08019444520415517539609438874, 8.506488033690131395685645540095, 9.274123664750092932637412448684, 10.22865745785398871442772506602, 11.02690930373063130412299282091, 12.65326923199955740394956527666, 13.84764416689358537674727777396
