L(s) = 1 | + (−1.5 − 0.866i)5-s + (0.5 − 2.59i)7-s + (−1.5 + 0.866i)11-s + (−3 + 1.73i)17-s + (1 − 1.73i)19-s + (−1 − 1.73i)25-s − 9·29-s + (−2.5 − 4.33i)31-s + (−3 + 3.46i)35-s + (−5 + 8.66i)37-s + 10.3i·41-s + 3.46i·43-s + (6 − 10.3i)47-s + (−6.5 − 2.59i)49-s + (−4.5 − 7.79i)53-s + ⋯ |
L(s) = 1 | + (−0.670 − 0.387i)5-s + (0.188 − 0.981i)7-s + (−0.452 + 0.261i)11-s + (−0.727 + 0.420i)17-s + (0.229 − 0.397i)19-s + (−0.200 − 0.346i)25-s − 1.67·29-s + (−0.449 − 0.777i)31-s + (−0.507 + 0.585i)35-s + (−0.821 + 1.42i)37-s + 1.62i·41-s + 0.528i·43-s + (0.875 − 1.51i)47-s + (−0.928 − 0.371i)49-s + (−0.618 − 1.07i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4387126835\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4387126835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (1.5 + 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12 - 6.92i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 + (6 - 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 1.73i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
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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
−9.648379453631351592936606508207, −8.623437817106801739532009345320, −7.85402883432371423855138908813, −7.23406371073138902944668589510, −6.24437818555985477889632961342, −4.97978908789291526217131863315, −4.29048662291944061885694971610, −3.35970825456074349191837411546, −1.79801334137394120496675071308, −0.19068555869781792360466605670,
1.97355284828027452590557258020, 3.08870515957561331948077713993, 4.07643661437941303177221755672, 5.31356309469058832063843697828, 5.92891124247341074304070858059, 7.25115737345175603387913699904, 7.67899035872686296459908945376, 8.912671000369812131225293491379, 9.196584443765097739552033397841, 10.63003244423538236457362486274