L(s) = 1 | + (−10.9 − 18.9i)5-s + (−12.1 − 13.9i)7-s + (−45.1 − 26.0i)11-s − 54.9i·13-s + (−40.8 + 70.7i)17-s + (113. − 65.6i)19-s + (−38.0 + 21.9i)23-s + (−176. + 305. i)25-s − 238. i·29-s + (174. + 100. i)31-s + (−132. + 382. i)35-s + (12.0 + 20.8i)37-s + 102.·41-s + 119.·43-s + (−20.2 − 35.1i)47-s + ⋯ |
L(s) = 1 | + (−0.977 − 1.69i)5-s + (−0.655 − 0.755i)7-s + (−1.23 − 0.714i)11-s − 1.17i·13-s + (−0.582 + 1.00i)17-s + (1.37 − 0.792i)19-s + (−0.345 + 0.199i)23-s + (−1.40 + 2.44i)25-s − 1.52i·29-s + (1.00 + 0.582i)31-s + (−0.638 + 1.84i)35-s + (0.0534 + 0.0925i)37-s + 0.389·41-s + 0.424·43-s + (−0.0629 − 0.109i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4783558539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4783558539\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (12.1 + 13.9i)T \) |
good | 5 | \( 1 + (10.9 + 18.9i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (45.1 + 26.0i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 54.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (40.8 - 70.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-113. + 65.6i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (38.0 - 21.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 238. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-174. - 100. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-12.0 - 20.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (20.2 + 35.1i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-297. - 172. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-142. + 246. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (386. - 223. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-113. + 196. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 886. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-6.46 - 3.73i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (404. + 700. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 943.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (575. + 996. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.55e3iT - 9.12e5T^{2} \) |
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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
−9.897172587085828915390749504482, −8.859714356626364372033037801185, −8.000480983495628686351342288009, −7.58393400379676839265086818276, −5.95562586174755076358092426786, −5.05480432992413963237287170337, −4.11003111286692554477234084087, −3.04812069813546936954760968725, −0.907618145121484527006637433688, −0.19193902966524359630876654648,
2.41716750996301781685291657151, 3.07541239950997763882505924100, 4.28801831250502242799768557856, 5.64090738052057706234623921351, 6.87162824344564442669025071143, 7.23457424404773084512900932921, 8.275403964970847323105016063397, 9.575791649973008440538936407053, 10.20146859289846688456452416497, 11.22942276396386625282368645825