L(s) = 1 | + (1.19 − 2.07i)5-s + 1.22i·7-s − 2.95i·11-s + (3 − 1.73i)13-s + (−0.138 + 0.239i)17-s + (−1.30 − 4.15i)19-s + (1.05 − 0.610i)23-s + (−0.361 − 0.626i)25-s + (6.58 − 3.80i)29-s − 8.90·31-s + (2.53 + 1.46i)35-s + 7.60i·37-s + (−8.26 − 4.77i)41-s + (−3 − 1.73i)43-s + (−4.05 + 2.34i)47-s + ⋯ |
L(s) = 1 | + (0.534 − 0.926i)5-s + 0.461i·7-s − 0.890i·11-s + (0.832 − 0.480i)13-s + (−0.0335 + 0.0581i)17-s + (−0.299 − 0.954i)19-s + (0.220 − 0.127i)23-s + (−0.0723 − 0.125i)25-s + (1.22 − 0.706i)29-s − 1.59·31-s + (0.427 + 0.246i)35-s + 1.25i·37-s + (−1.29 − 0.744i)41-s + (−0.457 − 0.264i)43-s + (−0.591 + 0.341i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.820151304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.820151304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.30 + 4.15i)T \) |
good | 5 | \( 1 + (-1.19 + 2.07i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 1.22iT - 7T^{2} \) |
| 11 | \( 1 + 2.95iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.138 - 0.239i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.05 + 0.610i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.58 + 3.80i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.90T + 31T^{2} \) |
| 37 | \( 1 - 7.60iT - 37T^{2} \) |
| 41 | \( 1 + (8.26 + 4.77i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3 + 1.73i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.05 - 2.34i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 3.46i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.44 + 5.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.19 + 5.53i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 - 6.64i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.75 + 3.03i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.392 - 0.679i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8.31iT - 83T^{2} \) |
| 89 | \( 1 + (-2.58 + 1.49i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.67 + 3.27i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.512037347351479889792549982354, −8.313391631907906794998178000925, −6.97276158364064849491331992014, −6.22709665372972562806473769459, −5.43421079954834796003153846962, −4.94236414029924611985725376911, −3.77865172058802319366260122464, −2.86106982173926784970592032526, −1.70141484667203465306243271462, −0.59840224288344752439814348413,
1.42765051488170276770502346583, 2.32548911636716198604692855038, 3.43593923365961204052229150795, 4.15643175774707877779518431188, 5.20869802249870071983580681143, 6.10481551060907681880305537204, 6.82529331592564238232333734896, 7.27771591660082701693506678439, 8.287540372014761083482473503899, 9.067979875177158729469453298036