L(s) = 1 | + (0.263 − 0.455i)5-s + (0.333 + 2.62i)7-s + (−2.30 − 3.99i)11-s + (0.244 + 0.423i)13-s + (−2.75 + 4.77i)17-s + (−1.83 − 3.18i)19-s + (0.0269 − 0.0467i)23-s + (2.36 + 4.09i)25-s + (3.28 − 5.68i)29-s − 6.07·31-s + (1.28 + 0.538i)35-s + (0.223 + 0.387i)37-s + (−2.52 − 4.36i)41-s + (−2.84 + 4.93i)43-s − 9.19·47-s + ⋯ |
L(s) = 1 | + (0.117 − 0.203i)5-s + (0.125 + 0.992i)7-s + (−0.695 − 1.20i)11-s + (0.0678 + 0.117i)13-s + (−0.668 + 1.15i)17-s + (−0.421 − 0.730i)19-s + (0.00562 − 0.00974i)23-s + (0.472 + 0.818i)25-s + (0.609 − 1.05i)29-s − 1.09·31-s + (0.216 + 0.0910i)35-s + (0.0367 + 0.0637i)37-s + (−0.394 − 0.682i)41-s + (−0.434 + 0.752i)43-s − 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09369616003\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09369616003\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.333 - 2.62i)T \) |
good | 5 | \( 1 + (-0.263 + 0.455i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.30 + 3.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.244 - 0.423i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.75 - 4.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.83 + 3.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0269 + 0.0467i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.28 + 5.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 6.07T + 31T^{2} \) |
| 37 | \( 1 + (-0.223 - 0.387i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.52 + 4.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.84 - 4.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.19T + 47T^{2} \) |
| 53 | \( 1 + (-4.37 + 7.57i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 + 0.465T + 61T^{2} \) |
| 67 | \( 1 + 5.19T + 67T^{2} \) |
| 71 | \( 1 - 1.76T + 71T^{2} \) |
| 73 | \( 1 + (5.23 - 9.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + (-4.49 + 7.78i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.05 - 12.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.22 + 9.04i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.604226577257986711429364283228, −7.82508853720076512894804419141, −6.70797705603573740790797410188, −6.01977586171362523420988742842, −5.38960417309694552848297312650, −4.57834861727634465686971396363, −3.46908288531931645864052440070, −2.61339596363219616585378748239, −1.64555759343829435786021291020, −0.02753599563080778102094712990,
1.49931849502870164101229604812, 2.53357355407116595555680485350, 3.51893305466384776013062133343, 4.60990713382670062936653048210, 4.94360553940222921179209263648, 6.17092818600782401391666881252, 7.02195027828143267037847938351, 7.38998278398079014012346891151, 8.244645628465707908323680914615, 9.096775320873209117478100916292