| L(s) = 1 | + (−0.285 + 2.26i)2-s + (1.43 − 1.19i)3-s + (−3.09 − 0.794i)4-s + (2.23 + 0.103i)5-s + (2.28 + 3.59i)6-s + (−1.01 + 0.330i)7-s + (1.00 − 2.53i)8-s + (0.0918 − 0.481i)9-s + (−0.871 + 5.02i)10-s + (1.56 + 0.197i)11-s + (−5.40 + 2.54i)12-s + (−3.41 − 0.651i)13-s + (−0.456 − 2.39i)14-s + (3.33 − 2.51i)15-s + (−0.154 − 0.0848i)16-s + (−1.57 − 6.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.202 + 1.59i)2-s + (0.831 − 0.687i)3-s + (−1.54 − 0.397i)4-s + (0.998 + 0.0462i)5-s + (0.931 + 1.46i)6-s + (−0.384 + 0.124i)7-s + (0.354 − 0.896i)8-s + (0.0306 − 0.160i)9-s + (−0.275 + 1.58i)10-s + (0.470 + 0.0594i)11-s + (−1.55 + 0.733i)12-s + (−0.947 − 0.180i)13-s + (−0.122 − 0.639i)14-s + (0.861 − 0.648i)15-s + (−0.0385 − 0.0212i)16-s + (−0.380 − 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.918263 + 0.829937i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.918263 + 0.829937i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.23 - 0.103i)T \) |
| good | 2 | \( 1 + (0.285 - 2.26i)T + (-1.93 - 0.497i)T^{2} \) |
| 3 | \( 1 + (-1.43 + 1.19i)T + (0.562 - 2.94i)T^{2} \) |
| 7 | \( 1 + (1.01 - 0.330i)T + (5.66 - 4.11i)T^{2} \) |
| 11 | \( 1 + (-1.56 - 0.197i)T + (10.6 + 2.73i)T^{2} \) |
| 13 | \( 1 + (3.41 + 0.651i)T + (12.0 + 4.78i)T^{2} \) |
| 17 | \( 1 + (1.57 + 6.11i)T + (-14.8 + 8.18i)T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.854i)T + (-3.56 - 18.6i)T^{2} \) |
| 23 | \( 1 + (5.21 - 5.55i)T + (-1.44 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.566 + 9.00i)T + (-28.7 + 3.63i)T^{2} \) |
| 31 | \( 1 + (-3.33 + 0.855i)T + (27.1 - 14.9i)T^{2} \) |
| 37 | \( 1 + (3.04 - 5.53i)T + (-19.8 - 31.2i)T^{2} \) |
| 41 | \( 1 + (-2.67 + 2.51i)T + (2.57 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-2.82 - 3.88i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (2.28 + 5.77i)T + (-34.2 + 32.1i)T^{2} \) |
| 53 | \( 1 + (4.97 + 3.16i)T + (22.5 + 47.9i)T^{2} \) |
| 59 | \( 1 + (-4.73 - 10.0i)T + (-37.6 + 45.4i)T^{2} \) |
| 61 | \( 1 + (0.277 + 0.261i)T + (3.83 + 60.8i)T^{2} \) |
| 67 | \( 1 + (13.2 + 0.831i)T + (66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-7.25 + 2.87i)T + (51.7 - 48.6i)T^{2} \) |
| 73 | \( 1 + (0.650 + 0.305i)T + (46.5 + 56.2i)T^{2} \) |
| 79 | \( 1 + (-4.42 - 5.34i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (-5.71 - 4.73i)T + (15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (1.50 - 3.19i)T + (-56.7 - 68.5i)T^{2} \) |
| 97 | \( 1 + (-13.1 + 0.830i)T + (96.2 - 12.1i)T^{2} \) |
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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
−13.73768608066929589145783118798, −13.39789689093651449848956162902, −11.84685688369607093838823481527, −9.769395082912383290710862137523, −9.229673825070936195743406419372, −7.985566293484577159362286155534, −7.16723370505623133764527368176, −6.20135140821714303897575877196, −5.00055153003937777380975554512, −2.49633513313733624933778146201,
2.01107511470895552346941018313, 3.28743462060528939479315581921, 4.45427355057888416178262549604, 6.40028977125862455731486442636, 8.560654177287997926604876404497, 9.319858193461532599244445027803, 10.07626613372759208405066868717, 10.70610355763468788991632456035, 12.26133204651657163972623272381, 12.84831009534021272842242428132
