L(s) = 1 | + (0.455 − 1.69i)2-s + (−1.67 − 0.421i)3-s + (−0.948 − 0.547i)4-s + (−0.445 + 2.19i)5-s + (−1.48 + 2.66i)6-s + (1.12 − 1.12i)8-s + (2.64 + 1.41i)9-s + (3.52 + 1.75i)10-s + (−1.34 − 0.776i)11-s + (1.36 + 1.32i)12-s + (−4.50 − 4.50i)13-s + (1.67 − 3.49i)15-s + (−2.49 − 4.32i)16-s + (−2.91 + 0.780i)17-s + (3.61 − 3.84i)18-s + (3.64 − 2.10i)19-s + ⋯ |
L(s) = 1 | + (0.322 − 1.20i)2-s + (−0.969 − 0.243i)3-s + (−0.474 − 0.273i)4-s + (−0.199 + 0.979i)5-s + (−0.604 + 1.08i)6-s + (0.397 − 0.397i)8-s + (0.881 + 0.472i)9-s + (1.11 + 0.554i)10-s + (−0.405 − 0.234i)11-s + (0.393 + 0.381i)12-s + (−1.25 − 1.25i)13-s + (0.431 − 0.902i)15-s + (−0.623 − 1.08i)16-s + (−0.706 + 0.189i)17-s + (0.851 − 0.907i)18-s + (0.836 − 0.482i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.114242 + 0.646845i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.114242 + 0.646845i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.67 + 0.421i)T \) |
| 5 | \( 1 + (0.445 - 2.19i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.455 + 1.69i)T + (-1.73 - i)T^{2} \) |
| 11 | \( 1 + (1.34 + 0.776i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.50 + 4.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.91 - 0.780i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.64 + 2.10i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.13 + 1.37i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 2.97T + 29T^{2} \) |
| 31 | \( 1 + (-2.89 + 5.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.68 - 0.450i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.68iT - 41T^{2} \) |
| 43 | \( 1 + (2.09 + 2.09i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.0131 + 0.0489i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.57 + 5.88i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.46 - 4.27i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.65 + 2.87i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.625 - 2.33i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 5.73iT - 71T^{2} \) |
| 73 | \( 1 + (9.92 - 2.66i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.11 + 1.79i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.2 + 12.2i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.678 + 1.17i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \) |
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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
−10.24141966503050226062000238632, −9.729901518120767485899790006717, −7.85496950837262841870526369826, −7.31391991650653008471214195373, −6.29357229210852370819168301222, −5.23740946718799072756748450842, −4.23262832276478750424368042282, −3.02157942412758923618187265040, −2.15824902481265305286640816532, −0.32547226403071567440692103503,
1.80531034800613146725769801685, 4.15834835692831738468740070594, 4.81616501336120718195328994074, 5.44435398026577302184794267845, 6.39760590515115331544083452961, 7.24878773259859862209460122612, 7.922954373496735866846032293195, 9.151113399793855250990329201003, 9.829538222055032510737983401612, 10.93051962863521431807579409586