L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−2.44 − 1.41i)5-s + (−1.5 + 0.866i)7-s − 2.82i·8-s + (1.99 + 3.46i)10-s + (2.44 + 4.24i)11-s + (0.5 − 0.866i)13-s + 2.44·14-s + (−2.00 + 3.46i)16-s + 2.82i·17-s + 5.19i·19-s − 5.65i·20-s − 6.92i·22-s + (−2.44 + 4.24i)23-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s + (−1.09 − 0.632i)5-s + (−0.566 + 0.327i)7-s − 0.999i·8-s + (0.632 + 1.09i)10-s + (0.738 + 1.27i)11-s + (0.138 − 0.240i)13-s + 0.654·14-s + (−0.500 + 0.866i)16-s + 0.685i·17-s + 1.19i·19-s − 1.26i·20-s − 1.47i·22-s + (−0.510 + 0.884i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.394274 + 0.276073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.394274 + 0.276073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.44 + 1.41i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + (2.44 - 4.24i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.89 - 2.82i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 - 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (-4.89 - 2.82i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3 + 1.73i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.44 - 4.24i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (2.44 - 4.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.5 + 6.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.89 + 8.48i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.82iT - 89T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.01126620412463733447554604782, −10.83514444275068475779660875203, −9.793289698365595241125676606392, −9.101975160900623508751713420915, −8.059651741492884359333396808855, −7.40717791477830515842155558723, −6.13904444905258672662918632568, −4.33234462581163810232217985443, −3.46857718759297321924094406682, −1.62713609108648205787100399277,
0.46586940799749692397480439633, 2.88611052024228313022310259255, 4.19844623681412881947878476658, 5.92466029761046441513729123120, 6.81860681185601038923189947835, 7.54697307992944941050436848352, 8.607503621460782449718296118875, 9.376365904650975442616030787927, 10.56509422090054354436415682461, 11.29590580015308143803140230585