L(s) = 1 | + (−0.298 + 0.0799i)2-s + (1.15 + 1.29i)3-s + (−1.64 + 0.952i)4-s + (1.56 − 1.59i)5-s + (−0.447 − 0.292i)6-s + (0.951 + 2.46i)7-s + (0.852 − 0.852i)8-s + (−0.333 + 2.98i)9-s + (−0.340 + 0.600i)10-s + (−0.660 + 0.381i)11-s + (−3.13 − 1.02i)12-s + (−2.27 − 2.27i)13-s + (−0.481 − 0.660i)14-s + (3.86 + 0.184i)15-s + (1.71 − 2.97i)16-s + (1.25 − 4.69i)17-s + ⋯ |
L(s) = 1 | + (−0.210 + 0.0565i)2-s + (0.666 + 0.745i)3-s + (−0.824 + 0.476i)4-s + (0.701 − 0.712i)5-s + (−0.182 − 0.119i)6-s + (0.359 + 0.933i)7-s + (0.301 − 0.301i)8-s + (−0.111 + 0.993i)9-s + (−0.107 + 0.189i)10-s + (−0.199 + 0.114i)11-s + (−0.904 − 0.297i)12-s + (−0.629 − 0.629i)13-s + (−0.128 − 0.176i)14-s + (0.998 + 0.0476i)15-s + (0.429 − 0.744i)16-s + (0.305 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.951699 + 0.446543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.951699 + 0.446543i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.15 - 1.29i)T \) |
| 5 | \( 1 + (-1.56 + 1.59i)T \) |
| 7 | \( 1 + (-0.951 - 2.46i)T \) |
good | 2 | \( 1 + (0.298 - 0.0799i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (0.660 - 0.381i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.27 + 2.27i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.25 + 4.69i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 0.818i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.98 + 7.39i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 + (-2.96 - 5.13i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.915 + 3.41i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 4.35iT - 41T^{2} \) |
| 43 | \( 1 + (-2.69 - 2.69i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.14 - 1.10i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.71 - 1.79i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.84 - 6.65i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0471 + 0.0126i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 + (-0.359 + 1.34i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.66 + 2.11i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.05 - 5.05i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.453 - 0.785i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.73 + 3.73i)T - 97iT^{2} \) |
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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
−13.99920429596656099167880159566, −12.96059429743635006630608208119, −12.07605849668141608362615803380, −10.27256839377607404031674377481, −9.422852579145509056349884036700, −8.697365105452968453904277484431, −7.78631863216822720128372315683, −5.40212568551258648384813353834, −4.61848967732270377794003124876, −2.72083510051971430192810921346,
1.74454804715826243937081266219, 3.81663674748601830592254861361, 5.67740741984274556196796251508, 7.07133504332943469769667803083, 8.087691135656367954286586490834, 9.447252873630185922125345976477, 10.14863833897985606208511081011, 11.42197141530771500397324141210, 13.05699419930619686874248991217, 13.71666647681976269727169916555