L(s) = 1 | + (0.5 − 0.866i)3-s + (−1 − 1.73i)5-s + (−0.499 − 0.866i)9-s + (−2 + 3.46i)11-s − 6·13-s − 1.99·15-s + (1 − 1.73i)17-s + (2 + 3.46i)19-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s − 0.999·27-s − 2·29-s + (1.99 + 3.46i)33-s + (5 + 8.66i)37-s + (−3 + 5.19i)39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.166 − 0.288i)9-s + (−0.603 + 1.04i)11-s − 1.66·13-s − 0.516·15-s + (0.242 − 0.420i)17-s + (0.458 + 0.794i)19-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s − 0.192·27-s − 0.371·29-s + (0.348 + 0.603i)33-s + (0.821 + 1.42i)37-s + (−0.480 + 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.137607217\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.137607217\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-5 + 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.200804559105473482135868930439, −8.076244317429636452184966764889, −7.52153575978162829343003240627, −7.18050712820144844023193820702, −5.86830990357849689029674012713, −4.98334870028065082320252117225, −4.47403221169126909865329529376, −3.18365227758415260160442981019, −2.28028841945653173515630096770, −1.10605447129512870137807243633,
0.41174382376545571868007050685, 2.51148999790816934865375500343, 2.92269533699439408689146854932, 3.96054539470267316462527588891, 4.89314829997180671735298956138, 5.62019350693438513105640873666, 6.71624139563978578411400443145, 7.43329317485650032101179761406, 8.032809544339703442378927911390, 8.965161309882261285383545758814