L(s) = 1 | + (1.45 − 2.51i)5-s + (−2.64 + 0.0146i)7-s + (1.08 − 0.628i)11-s − 5.32i·13-s + (0.880 + 1.52i)17-s + (−6.69 − 3.86i)19-s + (4.43 + 2.55i)23-s + (−1.72 − 2.98i)25-s − 8.74i·29-s + (2.18 − 1.26i)31-s + (−3.80 + 6.68i)35-s + (−3.66 + 6.34i)37-s − 3.09·41-s + 1.15·43-s + (3.44 − 5.96i)47-s + ⋯ |
L(s) = 1 | + (0.649 − 1.12i)5-s + (−0.999 + 0.00554i)7-s + (0.328 − 0.189i)11-s − 1.47i·13-s + (0.213 + 0.369i)17-s + (−1.53 − 0.887i)19-s + (0.924 + 0.533i)23-s + (−0.344 − 0.597i)25-s − 1.62i·29-s + (0.392 − 0.226i)31-s + (−0.643 + 1.12i)35-s + (−0.602 + 1.04i)37-s − 0.483·41-s + 0.176·43-s + (0.502 − 0.869i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0293 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0293 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.892546 - 0.919106i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.892546 - 0.919106i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.64 - 0.0146i)T \) |
good | 5 | \( 1 + (-1.45 + 2.51i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.08 + 0.628i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5.32iT - 13T^{2} \) |
| 17 | \( 1 + (-0.880 - 1.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.69 + 3.86i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.43 - 2.55i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8.74iT - 29T^{2} \) |
| 31 | \( 1 + (-2.18 + 1.26i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.66 - 6.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 + (-3.44 + 5.96i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 6.52i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.13 - 5.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.19 - 1.26i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.689 + 1.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.65iT - 71T^{2} \) |
| 73 | \( 1 + (-4.38 + 2.52i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.63 - 13.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 3.75T + 83T^{2} \) |
| 89 | \( 1 + (3.55 - 6.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 4.03iT - 97T^{2} \) |
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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
−10.43567902446626306838756326164, −9.836079764923831614168418883794, −8.873214331820329489504707726891, −8.287611783210496587901303847267, −6.89117964406228819507433198636, −5.93775589507254417647810365686, −5.15659053387679424659109215423, −3.88294340044373827136122333902, −2.53716139339846182045984488306, −0.76607945082392548398127674610,
2.01426795912849488728166649496, 3.15752703216957759103518315394, 4.30173088470062682166011240805, 5.83429543490501337503361858536, 6.75023042740233069267526472337, 7.01670923816524605597139866337, 8.736373645735561904474701980350, 9.398130079481021961736005841739, 10.38446757162985283336187891271, 10.84607174792085040469753750218