L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (1.5 − 2.59i)7-s + (1 − 1.73i)9-s + (2.5 − 4.33i)13-s + (−0.499 + 0.866i)15-s + (−1.5 + 2.59i)17-s + (0.5 + 0.866i)19-s + 3·21-s + (−3.5 − 6.06i)23-s + (2 − 3.46i)25-s + 5·27-s + (−1.5 + 2.59i)29-s + (2.5 + 4.33i)31-s + 3·35-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.566 − 0.981i)7-s + (0.333 − 0.577i)9-s + (0.693 − 1.20i)13-s + (−0.129 + 0.223i)15-s + (−0.363 + 0.630i)17-s + (0.114 + 0.198i)19-s + 0.654·21-s + (−0.729 − 1.26i)23-s + (0.400 − 0.692i)25-s + 0.962·27-s + (−0.278 + 0.482i)29-s + (0.449 + 0.777i)31-s + 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.88751 - 0.163164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.88751 - 0.163164i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 + (-4 - 5.19i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.50394288321461305729714503499, −9.836442806056808110997380227951, −8.586794310775862755486423803224, −8.039889548999201075403027815518, −6.84630890041493936075302620159, −6.11242632792189946982289547879, −4.72839371823650277054352650689, −3.92527711155799805691515586164, −2.90669861799568570656320516641, −1.14482069443833873669511896856,
1.59879615059605037345712103737, 2.39507147899462068171661761233, 4.04726142858671480018569921069, 5.11700664022436826688220904845, 5.95506543807325213765945792119, 7.13130282015602544162713464820, 7.88222145262658277150844246147, 8.922763163572864049796116384908, 9.265678116799833646400514511573, 10.54367726674060662275796009401