| L(s) = 1 | + (2.78 + 0.491i)2-s + (−2.20 + 2.63i)3-s + (7.51 + 2.73i)4-s + (27.1 − 9.88i)5-s + (−7.44 + 6.24i)6-s + (21.0 + 36.4i)7-s + (19.5 + 11.3i)8-s + (12.0 + 68.1i)9-s + (80.5 − 14.1i)10-s + (17.9 − 31.0i)11-s + (−23.8 + 13.7i)12-s + (−193. − 230. i)13-s + (40.7 + 111. i)14-s + (−33.9 + 93.3i)15-s + (49.0 + 41.1i)16-s + (61.5 − 348. i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (−0.245 + 0.292i)3-s + (0.469 + 0.171i)4-s + (1.08 − 0.395i)5-s + (−0.206 + 0.173i)6-s + (0.429 + 0.744i)7-s + (0.306 + 0.176i)8-s + (0.148 + 0.841i)9-s + (0.805 − 0.141i)10-s + (0.148 − 0.256i)11-s + (−0.165 + 0.0954i)12-s + (−1.14 − 1.36i)13-s + (0.207 + 0.571i)14-s + (−0.150 + 0.414i)15-s + (0.191 + 0.160i)16-s + (0.212 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.19296 + 0.535439i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.19296 + 0.535439i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.78 - 0.491i)T \) |
| 19 | \( 1 + (316. - 173. i)T \) |
| good | 3 | \( 1 + (2.20 - 2.63i)T + (-14.0 - 79.7i)T^{2} \) |
| 5 | \( 1 + (-27.1 + 9.88i)T + (478. - 401. i)T^{2} \) |
| 7 | \( 1 + (-21.0 - 36.4i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-17.9 + 31.0i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (193. + 230. i)T + (-4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-61.5 + 348. i)T + (-7.84e4 - 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-39.9 - 14.5i)T + (2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (989. - 174. i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-483. + 278. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 598. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-1.10e3 + 1.31e3i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-749. + 272. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-649. - 3.68e3i)T + (-4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-463. + 1.27e3i)T + (-6.04e6 - 5.07e6i)T^{2} \) |
| 59 | \( 1 + (4.58e3 + 808. i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (1.89e3 + 689. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-5.81e3 + 1.02e3i)T + (1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-1.39e3 - 3.84e3i)T + (-1.94e7 + 1.63e7i)T^{2} \) |
| 73 | \( 1 + (5.26e3 + 4.42e3i)T + (4.93e6 + 2.79e7i)T^{2} \) |
| 79 | \( 1 + (-996. + 1.18e3i)T + (-6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-5.50e3 - 9.54e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-5.00e3 - 5.97e3i)T + (-1.08e7 + 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-1.59e4 - 2.80e3i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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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
−15.52561113677626495333748171381, −14.41044883390025702904732035468, −13.28935071012687585881320517468, −12.26396537264278012799352130965, −10.79808179511934594039700042417, −9.524005621787769885859332140969, −7.78993509274263712933813756730, −5.73019865599685244949886329843, −4.98965476100263205136678020064, −2.36138113314328916279969380310,
1.90372387092495166640478280439, 4.27634418827802748586521384978, 6.12344085283909302191235673620, 7.13090829555554349099843199382, 9.453299454364169327953822861826, 10.66198499458289804888020519737, 12.00799393462798056863261925078, 13.14274945477565037378189914380, 14.28918847728802732047438976795, 14.95613421664380220462543123791
