L(s) = 1 | + (10.1 − 17.5i)5-s + (−15.4 + 8.92i)11-s + 33.1i·13-s + (22.9 + 39.6i)17-s + (35.0 + 20.2i)19-s + (−69.7 − 40.2i)23-s + (−142. − 246. i)25-s + 233. i·29-s + (195. − 112. i)31-s + (135. − 234. i)37-s + 154.·41-s + 367.·43-s + (263. − 457. i)47-s + (78.8 − 45.5i)53-s + 361. i·55-s + ⋯ |
L(s) = 1 | + (0.905 − 1.56i)5-s + (−0.423 + 0.244i)11-s + 0.707i·13-s + (0.326 + 0.566i)17-s + (0.422 + 0.244i)19-s + (−0.632 − 0.365i)23-s + (−1.14 − 1.97i)25-s + 1.49i·29-s + (1.13 − 0.653i)31-s + (0.601 − 1.04i)37-s + 0.588·41-s + 1.30·43-s + (0.818 − 1.41i)47-s + (0.204 − 0.118i)53-s + 0.885i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.536132103\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.536132103\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (-10.1 + 17.5i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (15.4 - 8.92i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 33.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-22.9 - 39.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-35.0 - 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (69.7 + 40.2i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 233. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-195. + 112. i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-135. + 234. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-263. + 457. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-78.8 + 45.5i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-312. - 541. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-78.8 - 45.5i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (431. + 747. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 303. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-999. + 576. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-3.48 + 6.02i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 815.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (155. - 269. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.83e3iT - 9.12e5T^{2} \) |
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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
−8.829095516320580420115139195911, −8.165040891729598469801618529212, −7.24376193477278766763130228555, −6.06787535854593264035217720196, −5.55961137318818211891696784128, −4.68452023859426951235276744819, −3.97798666593167869009155127639, −2.43154849231270377877042980022, −1.58242251634078591076344351054, −0.61039354121528913961778899173,
0.999201223581665633324522601299, 2.58484392660914460287492485303, 2.76311992329447726672086747905, 3.98146391503147429171317194408, 5.30669256036148982807969005889, 5.97190317804897598178689489527, 6.63256463464060364224410818000, 7.54181718474328664361767090785, 8.097058001854701031319466925491, 9.442355623211761731480867593449