L(s) = 1 | + 3-s − 5-s + (−0.5 + 0.866i)7-s + 9-s + (−1.5 + 2.59i)11-s + (0.5 + 0.866i)13-s − 15-s + (−0.5 − 0.866i)17-s + (2.5 + 4.33i)19-s + (−0.5 + 0.866i)21-s + (−0.5 − 0.866i)23-s + 25-s + 27-s + (−2.5 + 4.33i)29-s + (3.5 − 6.06i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + (−0.188 + 0.327i)7-s + 0.333·9-s + (−0.452 + 0.783i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (−0.121 − 0.210i)17-s + (0.573 + 0.993i)19-s + (−0.109 + 0.188i)21-s + (−0.104 − 0.180i)23-s + 0.200·25-s + 0.192·27-s + (−0.464 + 0.804i)29-s + (0.628 − 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9767266752\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9767266752\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (8 + 1.73i)T \) |
good | 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.5 + 6.06i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (7.5 - 12.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.5 - 6.06i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.763948446136853868915146012502, −7.88015957608760105202312742878, −7.49361902728862253042698490210, −6.65614115853962734514606116445, −5.76781972019159061288266116323, −4.90183488400659615006908133477, −4.10309759386509739502640804763, −3.30731288219043116313833188952, −2.43487779921722442652741851143, −1.46506103660974575669545090414,
0.25221951214342277236937882900, 1.52084185058641043649189283630, 2.87427175041826401378324186435, 3.30639192611725997329959207048, 4.26834609183282095216384073222, 5.07377377997139603818937680317, 5.95135495486022931828543595627, 6.89851767334908217496606542213, 7.41515960356270389460315292879, 8.324667158009391933883234380201