L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.939 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)6-s + 0.999·8-s + (−1.26 + 2.19i)9-s + 0.347·11-s − 1.87·12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 2.53·18-s + (−0.173 − 0.300i)22-s + (0.939 + 1.62i)24-s + (−0.5 + 0.866i)25-s − 2.87·27-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.939 + 1.62i)3-s + (−0.499 + 0.866i)4-s + (0.939 − 1.62i)6-s + 0.999·8-s + (−1.26 + 2.19i)9-s + 0.347·11-s − 1.87·12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + 2.53·18-s + (−0.173 − 0.300i)22-s + (0.939 + 1.62i)24-s + (−0.5 + 0.866i)25-s − 2.87·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0238 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.182315215\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182315215\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.347T + T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - 1.53T + T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.181104827217375139045018642953, −8.734244492170545490508121988999, −8.033848983258140730640090005897, −7.30826158546439051407038919307, −5.78422545339988296104524075136, −4.90559113488590231640626798433, −4.03723646811873972787289549516, −3.59732534699859407151847611648, −2.76738339141353640070273612832, −1.74901778776636730931783788250,
0.810469590793909577491085084232, 1.82621902203459743614142187765, 2.80825583896444296002341311009, 3.95307650904102647586440589977, 5.20813550873666993337407558325, 6.17042089850362100298032707037, 6.65567159223329309352848942021, 7.41960632250225210865092661947, 7.87434131349372631516717230213, 8.541883933848905682685522657645