L(s) = 1 | + (0.5 + 0.866i)3-s + (1.45 + 0.842i)5-s + (−0.499 + 0.866i)9-s + (3.05 − 1.76i)11-s − 2.93i·13-s + 1.68i·15-s + (1.74 − 1.00i)17-s + (0.848 − 1.47i)19-s + (1.38 + 0.799i)23-s + (−1.08 − 1.87i)25-s − 0.999·27-s + 7.94·29-s + (−2.47 − 4.29i)31-s + (3.05 + 1.76i)33-s + (5.23 − 9.06i)37-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.652 + 0.376i)5-s + (−0.166 + 0.288i)9-s + (0.922 − 0.532i)11-s − 0.812i·13-s + 0.434i·15-s + (0.422 − 0.243i)17-s + (0.194 − 0.337i)19-s + (0.288 + 0.166i)23-s + (−0.216 − 0.374i)25-s − 0.192·27-s + 1.47·29-s + (−0.444 − 0.770i)31-s + (0.532 + 0.307i)33-s + (0.860 − 1.49i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.459238865\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.459238865\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.45 - 0.842i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.05 + 1.76i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + (-1.74 + 1.00i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.848 + 1.47i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.38 - 0.799i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 + (2.47 + 4.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.23 + 9.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.86iT - 41T^{2} \) |
| 43 | \( 1 - 11.7iT - 43T^{2} \) |
| 47 | \( 1 + (3.35 - 5.80i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.46 + 2.52i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.87 + 6.71i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.9 - 6.29i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.47 + 0.850i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.13iT - 71T^{2} \) |
| 73 | \( 1 + (2.10 - 1.21i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.749 + 0.432i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + (-13.8 - 7.96i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.82iT - 97T^{2} \) |
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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
−9.199429501768806427471635664312, −8.262271687425562172935560540251, −7.58788697575771467933850514030, −6.49813653765910442936298116054, −5.94094273292088288383559635431, −5.06220617777029044760860438203, −4.08352553095352351957523884277, −3.17048534572636319076338147557, −2.40199786335123758772719899022, −0.954423420583798568274496786683,
1.22152352125500906209760036755, 1.90192837977972710649925783370, 3.10200734398998696188461307192, 4.13672083316022655769634493551, 5.01807573851579208955994278336, 5.96850813886157667714819094604, 6.71750220879687943584201142680, 7.27518336337922621536502543643, 8.414972825609088826059945867406, 8.836634525051546133926778355099