L(s) = 1 | + (−2.12 − 0.707i)5-s + (−3.46 − 3.46i)7-s − 4.89i·11-s + (−1 + i)13-s + (2.82 − 2.82i)17-s − 6.92i·19-s + (3.99 + 3i)25-s − 7.07·29-s + 6.92·31-s + (4.89 + 9.79i)35-s + (−1 − i)37-s − 9.89i·41-s + (−6.92 + 6.92i)43-s + (4.89 − 4.89i)47-s + 16.9i·49-s + ⋯ |
L(s) = 1 | + (−0.948 − 0.316i)5-s + (−1.30 − 1.30i)7-s − 1.47i·11-s + (−0.277 + 0.277i)13-s + (0.685 − 0.685i)17-s − 1.58i·19-s + (0.799 + 0.600i)25-s − 1.31·29-s + 1.24·31-s + (0.828 + 1.65i)35-s + (−0.164 − 0.164i)37-s − 1.54i·41-s + (−1.05 + 1.05i)43-s + (0.714 − 0.714i)47-s + 2.42i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5323080410\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5323080410\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.12 + 0.707i)T \) |
good | 7 | \( 1 + (3.46 + 3.46i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.89iT - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 7.07T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.89iT - 41T^{2} \) |
| 43 | \( 1 + (6.92 - 6.92i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.89 + 4.89i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (-6.92 - 6.92i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (7 - 7i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + (4.89 + 4.89i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + (-5 - 5i)T + 97iT^{2} \) |
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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
−8.329405557317505597194063383615, −7.32366127422649796934721081710, −7.05152001678981593732653158417, −6.12966848306719759123310548772, −5.11713702010957528685056690172, −4.19242243126562208120986932863, −3.46185112242296767910533152157, −2.89211040006300963868471797534, −0.855969000286244316813069816720, −0.22726961510567736278823297481,
1.79236955736606320461160543112, 2.88056195324428767634892765513, 3.55226660263694279875391489962, 4.46016487568409206716030709555, 5.52440432053725602108065382832, 6.21416569037109889975637914223, 6.96043902156046299685812838204, 7.80749122942940486235297568968, 8.337433266201270432281509922559, 9.364952285500054313590908108327