L(s) = 1 | + (2 + 3.46i)5-s + (0.5 − 0.866i)7-s + (1 − 1.73i)11-s + (3 + 5.19i)13-s + 4·17-s − 4·19-s + (1 + 1.73i)23-s + (−5.49 + 9.52i)25-s + (−1 + 1.73i)29-s + 3.99·35-s + 2·37-s + (2 − 3.46i)43-s + (6 − 10.3i)47-s + (−0.499 − 0.866i)49-s + 6·53-s + ⋯ |
L(s) = 1 | + (0.894 + 1.54i)5-s + (0.188 − 0.327i)7-s + (0.301 − 0.522i)11-s + (0.832 + 1.44i)13-s + 0.970·17-s − 0.917·19-s + (0.208 + 0.361i)23-s + (−1.09 + 1.90i)25-s + (−0.185 + 0.321i)29-s + 0.676·35-s + 0.328·37-s + (0.304 − 0.528i)43-s + (0.875 − 1.51i)47-s + (−0.0714 − 0.123i)49-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.269906393\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.269906393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 - 5.19i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2 - 3.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (-1 + 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.209244511652132367570938159353, −8.554336166548360714450914031972, −7.37438512137802505087350948349, −6.84325840043061817446141743172, −6.15743767837629245770597005984, −5.52918283539738611742987398832, −4.11546490233058554246421005037, −3.44651607307765028468195829273, −2.38997250007477830614322238785, −1.45908415495041964824082576872,
0.839206932354676888316227230424, 1.70777277755232575227514737674, 2.88681805925472337816199358023, 4.20454159034643615099707557631, 4.89992899591419201782855103922, 5.84609237264772079633283091261, 6.03942213963510533127224475885, 7.53253180884915475942636283112, 8.255776400714484062082401471138, 8.839182060317718036622328685070