L(s) = 1 | + (0.403 + 0.403i)2-s − 1.67i·4-s + (−1.03 + 1.03i)5-s + (2.60 + 0.450i)7-s + (1.48 − 1.48i)8-s − 0.832·10-s + (0.596 − 0.596i)11-s + (−3.59 + 0.296i)13-s + (0.869 + 1.23i)14-s − 2.15·16-s + 7.34·17-s + (3.59 − 3.59i)19-s + (1.72 + 1.72i)20-s + 0.481·22-s − 4.44i·23-s + ⋯ |
L(s) = 1 | + (0.284 + 0.284i)2-s − 0.837i·4-s + (−0.461 + 0.461i)5-s + (0.985 + 0.170i)7-s + (0.523 − 0.523i)8-s − 0.263·10-s + (0.179 − 0.179i)11-s + (−0.996 + 0.0822i)13-s + (0.232 + 0.329i)14-s − 0.539·16-s + 1.78·17-s + (0.824 − 0.824i)19-s + (0.386 + 0.386i)20-s + 0.102·22-s − 0.926i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86625 - 0.361697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86625 - 0.361697i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.60 - 0.450i)T \) |
| 13 | \( 1 + (3.59 - 0.296i)T \) |
good | 2 | \( 1 + (-0.403 - 0.403i)T + 2iT^{2} \) |
| 5 | \( 1 + (1.03 - 1.03i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.596 + 0.596i)T - 11iT^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + (-3.59 + 3.59i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.44iT - 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 + (1.27 - 1.27i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.88 + 2.88i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.23 - 1.23i)T - 41iT^{2} \) |
| 43 | \( 1 + 8.66iT - 43T^{2} \) |
| 47 | \( 1 + (2.52 + 2.52i)T + 47iT^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 + (-1.08 - 1.08i)T + 59iT^{2} \) |
| 61 | \( 1 + 7.10iT - 61T^{2} \) |
| 67 | \( 1 + (-8.76 - 8.76i)T + 67iT^{2} \) |
| 71 | \( 1 + (-1.46 - 1.46i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.103 + 0.103i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.79T + 79T^{2} \) |
| 83 | \( 1 + (12.3 - 12.3i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.89 - 6.89i)T + 89iT^{2} \) |
| 97 | \( 1 + (6.05 - 6.05i)T - 97iT^{2} \) |
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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
−10.20997275182794018780184346569, −9.483916908502757501714646579667, −8.365909670789663832632296015305, −7.40779782058585469192423543589, −6.86008058826017699258984076299, −5.49383432173288779158848781118, −5.09309589350309831534996655029, −3.92657096539607778129819863170, −2.54496383292511640867257176254, −1.04194100177051884101677690715,
1.39915914005391202519108058685, 2.87719295486086940904754701810, 3.91785179497408969937331340924, 4.76847446990993630555736292835, 5.56627668682546438297939059042, 7.21091336471180432218259686710, 7.87002128500694099256289952880, 8.232345882642598752433093852088, 9.511480819557359704318148240872, 10.29686948812920067483000947536