| L(s) = 1 | + (1.95 − 1.46i)3-s + (0.376 − 2.20i)5-s + (−0.0579 + 0.810i)7-s + (0.834 − 2.84i)9-s + (−0.249 + 0.387i)11-s + (3.31 − 0.236i)13-s + (−2.49 − 4.85i)15-s + (−0.626 + 1.68i)17-s + (1.93 − 4.23i)19-s + (1.07 + 1.66i)21-s + (−4.27 − 2.16i)23-s + (−4.71 − 1.65i)25-s + (0.0323 + 0.0868i)27-s + (−2.18 + 0.998i)29-s + (0.889 + 6.18i)31-s + ⋯ |
| L(s) = 1 | + (1.12 − 0.844i)3-s + (0.168 − 0.985i)5-s + (−0.0219 + 0.306i)7-s + (0.278 − 0.947i)9-s + (−0.0751 + 0.116i)11-s + (0.918 − 0.0657i)13-s + (−0.642 − 1.25i)15-s + (−0.151 + 0.407i)17-s + (0.443 − 0.972i)19-s + (0.234 + 0.364i)21-s + (−0.892 − 0.451i)23-s + (−0.943 − 0.331i)25-s + (0.00623 + 0.0167i)27-s + (−0.406 + 0.185i)29-s + (0.159 + 1.11i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65361 - 1.21620i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65361 - 1.21620i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.376 + 2.20i)T \) |
| 23 | \( 1 + (4.27 + 2.16i)T \) |
| good | 3 | \( 1 + (-1.95 + 1.46i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.0579 - 0.810i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (0.249 - 0.387i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.31 + 0.236i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (0.626 - 1.68i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 4.23i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.18 - 0.998i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.889 - 6.18i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (2.09 + 3.83i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (2.47 - 0.726i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.47 - 1.97i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (8.32 - 8.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.92 - 0.280i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-9.87 - 8.55i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-2.26 + 0.325i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-1.41 - 0.307i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (1.18 - 0.760i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-11.3 + 4.23i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (0.441 - 0.509i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-2.95 + 1.61i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (0.488 - 3.39i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (3.21 + 1.75i)T + (52.4 + 81.6i)T^{2} \) |
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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
−10.90789849603118155436574555437, −9.651832990804749439692539328279, −8.754676824621178533605680250176, −8.398378275852025683433031820819, −7.40469906809481515285207970691, −6.30999180797763043019488305082, −5.15044180323164703911900392418, −3.81910833507271402008608881060, −2.48983152810752254645521035355, −1.32399556094445454159196966453,
2.17940096343126598260134213682, 3.46043759804401210188390221010, 3.93729025894886093204027136630, 5.56132668450038241698068025200, 6.67773504053459712129036766008, 7.80865956263248124780004374665, 8.519962626086316346390301864258, 9.714711773184683286935902531163, 10.04096930137529766291702192077, 11.04773623012879713667426966178
