L(s) = 1 | + (−0.257 + 1.39i)2-s + (−1.66 + 1.66i)3-s + (−1.86 − 0.715i)4-s + (−0.707 − 0.707i)5-s + (−1.88 − 2.74i)6-s + 2.89i·7-s + (1.47 − 2.41i)8-s − 2.53i·9-s + (1.16 − 0.801i)10-s + (1.84 + 1.84i)11-s + (4.29 − 1.91i)12-s + (−3.08 + 3.08i)13-s + (−4.02 − 0.744i)14-s + 2.35·15-s + (2.97 + 2.67i)16-s + 7.29·17-s + ⋯ |
L(s) = 1 | + (−0.181 + 0.983i)2-s + (−0.960 + 0.960i)3-s + (−0.933 − 0.357i)4-s + (−0.316 − 0.316i)5-s + (−0.769 − 1.11i)6-s + 1.09i·7-s + (0.521 − 0.853i)8-s − 0.845i·9-s + (0.368 − 0.253i)10-s + (0.556 + 0.556i)11-s + (1.24 − 0.553i)12-s + (−0.854 + 0.854i)13-s + (−1.07 − 0.198i)14-s + 0.607·15-s + (0.744 + 0.667i)16-s + 1.77·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0928056 + 0.542761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0928056 + 0.542761i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.257 - 1.39i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (1.66 - 1.66i)T - 3iT^{2} \) |
| 7 | \( 1 - 2.89iT - 7T^{2} \) |
| 11 | \( 1 + (-1.84 - 1.84i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.08 - 3.08i)T - 13iT^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 + (1.23 - 1.23i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.60iT - 23T^{2} \) |
| 29 | \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.06T + 31T^{2} \) |
| 37 | \( 1 + (1.17 + 1.17i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.61iT - 41T^{2} \) |
| 43 | \( 1 + (-3.03 - 3.03i)T + 43iT^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + (-2.73 - 2.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \) |
| 61 | \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \) |
| 67 | \( 1 + (-8.24 + 8.24i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.25iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 0.113T + 79T^{2} \) |
| 83 | \( 1 + (-9.76 + 9.76i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.74iT - 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.04916599349838158583265460202, −14.37562126627049743189675909840, −12.45321322321811966929187599860, −11.77723480016359803312400669645, −10.10380359667736535198896798707, −9.403617648452073117310403902371, −8.069692462943350046187945433367, −6.43794282742513278656978756327, −5.29175108229792716368871042827, −4.35940089717309867763327357861,
0.931318203892303528770484283224, 3.47612250454271593007340283161, 5.34867073685228489067700649992, 7.04414505515548922092027906268, 8.009191967631877203615173692207, 9.902504592126109091657337519928, 10.82141117770155749271078176380, 11.81293069923267595563058990473, 12.50592131344513304143180107662, 13.57834604514510988001453272074