| L(s) = 1 | + (−0.690 + 1.23i)2-s + (2.07 + 2.47i)3-s + (−1.04 − 1.70i)4-s + (−2.16 + 0.568i)5-s + (−4.48 + 0.853i)6-s + (−1.40 + 2.44i)7-s + (2.82 − 0.113i)8-s + (−1.28 + 7.31i)9-s + (0.791 − 3.06i)10-s + (3.57 − 2.06i)11-s + (2.04 − 6.12i)12-s + (−0.432 − 0.363i)13-s + (−2.03 − 3.42i)14-s + (−5.89 − 4.16i)15-s + (−1.81 + 3.56i)16-s + (−6.16 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.488 + 0.872i)2-s + (1.19 + 1.42i)3-s + (−0.523 − 0.852i)4-s + (−0.967 + 0.254i)5-s + (−1.83 + 0.348i)6-s + (−0.532 + 0.922i)7-s + (0.999 − 0.0402i)8-s + (−0.429 + 2.43i)9-s + (0.250 − 0.968i)10-s + (1.07 − 0.622i)11-s + (0.590 − 1.76i)12-s + (−0.120 − 0.100i)13-s + (−0.544 − 0.915i)14-s + (−1.52 − 1.07i)15-s + (−0.452 + 0.891i)16-s + (−1.49 + 0.263i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.116584 - 1.09216i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116584 - 1.09216i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.690 - 1.23i)T \) |
| 5 | \( 1 + (2.16 - 0.568i)T \) |
| 19 | \( 1 + (-3.10 + 3.06i)T \) |
| good | 3 | \( 1 + (-2.07 - 2.47i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (1.40 - 2.44i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.57 + 2.06i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.432 + 0.363i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.16 - 1.08i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.24 + 0.451i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (2.13 + 0.376i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.48 - 4.30i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.45T + 37T^{2} \) |
| 41 | \( 1 + (-2.98 - 3.55i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-7.73 - 2.81i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.670 - 3.80i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (7.29 - 2.65i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.57 - 8.92i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-7.95 + 2.89i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-4.11 - 0.725i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.36 + 1.58i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.244 - 0.291i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.60 + 5.54i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.842 - 1.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.63 + 6.71i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.0553 + 0.313i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.36929395627153382906660474887, −10.76131438546602429517773266317, −9.467910764370308411172003080635, −9.056192370744110591683178137298, −8.475827595755244200195156814889, −7.41875272171606668616148123436, −6.19824611277096688196810004746, −4.81285453964253973155845640521, −3.94923659715978804420123762273, −2.78614067747703337776656714313,
0.78098008143989527472466556194, 2.11877820731923578068644714221, 3.51846996888275082319163365738, 4.15170495895178751178350180711, 6.80170433235948214075649764497, 7.30278877034685328987388510852, 8.077193520036143027760884035365, 9.061450423655140760482086727400, 9.584641911581104160979894250360, 11.12647441219845201677788212272
