L(s) = 1 | + (0.207 − 0.358i)3-s + (−1.62 − 2.09i)7-s + (1.41 + 2.44i)9-s + (−0.414 + 0.717i)11-s − 2·13-s + (−3.82 + 6.63i)17-s + (2.82 + 4.89i)19-s + (−1.08 + 0.148i)21-s + (−2.79 − 4.83i)23-s + 2.41·27-s − 7.82·29-s + (−0.414 + 0.717i)31-s + (0.171 + 0.297i)33-s + (2.82 + 4.89i)37-s + (−0.414 + 0.717i)39-s + ⋯ |
L(s) = 1 | + (0.119 − 0.207i)3-s + (−0.612 − 0.790i)7-s + (0.471 + 0.816i)9-s + (−0.124 + 0.216i)11-s − 0.554·13-s + (−0.928 + 1.60i)17-s + (0.648 + 1.12i)19-s + (−0.236 + 0.0324i)21-s + (−0.582 − 1.00i)23-s + 0.464·27-s − 1.45·29-s + (−0.0743 + 0.128i)31-s + (0.0298 + 0.0517i)33-s + (0.464 + 0.805i)37-s + (−0.0663 + 0.114i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.015780038\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015780038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.62 + 2.09i)T \) |
good | 3 | \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.414 - 0.717i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.79 + 4.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + (0.414 - 0.717i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + (-5.82 - 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.82 - 4.89i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.32 + 5.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.44 - 11.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-1.82 + 3.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 + (-2.67 - 4.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.893395574041003305068751563994, −9.018542534061322014711495645505, −7.84082619815839072373536256098, −7.58954143660168968696931818755, −6.51369718810926880530182727456, −5.79157157349672635529882034176, −4.47393608699894926136019936824, −3.93606414936728860722471152184, −2.57763522306340093920169065299, −1.49115930108142756793071327981,
0.39803953002408931483369339654, 2.27777779822076346478970234691, 3.13637838588076258290975611670, 4.17760411041282081733129406371, 5.22616166657849796455177735601, 5.98467260816422169940636112643, 7.06608519934883792704180485853, 7.49303348968151024107851414602, 9.053608552144786046255044478242, 9.205759219156756039035177310484