L(s) = 1 | − 1.02i·2-s + (−1.97 + 1.97i)3-s + 0.954·4-s + (−1.45 − 1.69i)5-s + (2.01 + 2.01i)6-s + 0.963·7-s − 3.02i·8-s − 4.76i·9-s + (−1.73 + 1.49i)10-s + (−1.17 + 1.17i)11-s + (−1.88 + 1.88i)12-s − 0.985i·14-s + (6.21 + 0.469i)15-s − 1.18·16-s + (−5.12 + 5.12i)17-s − 4.87·18-s + ⋯ |
L(s) = 1 | − 0.723i·2-s + (−1.13 + 1.13i)3-s + 0.477·4-s + (−0.651 − 0.758i)5-s + (0.822 + 0.822i)6-s + 0.364·7-s − 1.06i·8-s − 1.58i·9-s + (−0.548 + 0.471i)10-s + (−0.354 + 0.354i)11-s + (−0.542 + 0.542i)12-s − 0.263i·14-s + (1.60 + 0.121i)15-s − 0.295·16-s + (−1.24 + 1.24i)17-s − 1.14·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00141585 + 0.213150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00141585 + 0.213150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.45 + 1.69i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 1.02iT - 2T^{2} \) |
| 3 | \( 1 + (1.97 - 1.97i)T - 3iT^{2} \) |
| 7 | \( 1 - 0.963T + 7T^{2} \) |
| 11 | \( 1 + (1.17 - 1.17i)T - 11iT^{2} \) |
| 17 | \( 1 + (5.12 - 5.12i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.93 + 1.93i)T - 19iT^{2} \) |
| 23 | \( 1 + (2.72 + 2.72i)T + 23iT^{2} \) |
| 29 | \( 1 + 0.292iT - 29T^{2} \) |
| 31 | \( 1 + (0.125 + 0.125i)T + 31iT^{2} \) |
| 37 | \( 1 + 4.08T + 37T^{2} \) |
| 41 | \( 1 + (4.89 + 4.89i)T + 41iT^{2} \) |
| 43 | \( 1 + (5.62 + 5.62i)T + 43iT^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 53 | \( 1 + (1.99 - 1.99i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.57 - 3.57i)T + 59iT^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 - 7.29iT - 67T^{2} \) |
| 71 | \( 1 + (9.22 + 9.22i)T + 71iT^{2} \) |
| 73 | \( 1 + 3.22iT - 73T^{2} \) |
| 79 | \( 1 + 13.5iT - 79T^{2} \) |
| 83 | \( 1 + 8.56T + 83T^{2} \) |
| 89 | \( 1 + (0.366 + 0.366i)T + 89iT^{2} \) |
| 97 | \( 1 - 7.51iT - 97T^{2} \) |
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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
−10.23994849463994907843421649180, −9.179214110885652967231380758764, −8.276305452750931474902882156674, −7.04985750531035555246778886246, −6.09577923643428774782518839873, −5.02273412663192977759016574189, −4.35192127243358550343548620860, −3.52149404740169573274284977525, −1.81451616194069955192002821659, −0.11245504545881684419442684407,
1.77805904415179210057863097187, 3.02907651278299000243045591810, 4.81863203137735381048560388313, 5.65008519634064258343521606477, 6.61270012216570025631573975472, 6.93714119647226015065404439703, 7.77764772191212822831158234392, 8.336274295441259602038865971145, 9.990905476249663546863022336010, 11.14056455368171440607474333935