L(s) = 1 | + 1.19i·2-s − 2.62i·3-s + 0.580·4-s + (0.0644 + 2.23i)5-s + 3.12·6-s − 0.593i·7-s + 3.07i·8-s − 3.88·9-s + (−2.66 + 0.0767i)10-s − 11-s − 1.52i·12-s + 6.20i·13-s + 0.707·14-s + (5.86 − 0.169i)15-s − 2.50·16-s + 4.18i·17-s + ⋯ |
L(s) = 1 | + 0.842i·2-s − 1.51i·3-s + 0.290·4-s + (0.0288 + 0.999i)5-s + 1.27·6-s − 0.224i·7-s + 1.08i·8-s − 1.29·9-s + (−0.842 + 0.0242i)10-s − 0.301·11-s − 0.439i·12-s + 1.72i·13-s + 0.188·14-s + (1.51 − 0.0436i)15-s − 0.625·16-s + 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0288 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.572819633\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.572819633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.0644 - 2.23i)T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 1.19iT - 2T^{2} \) |
| 3 | \( 1 + 2.62iT - 3T^{2} \) |
| 7 | \( 1 + 0.593iT - 7T^{2} \) |
| 13 | \( 1 - 6.20iT - 13T^{2} \) |
| 17 | \( 1 - 4.18iT - 17T^{2} \) |
| 23 | \( 1 - 2.93iT - 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 - 1.18iT - 37T^{2} \) |
| 41 | \( 1 - 2.97T + 41T^{2} \) |
| 43 | \( 1 - 2.99iT - 43T^{2} \) |
| 47 | \( 1 + 7.94iT - 47T^{2} \) |
| 53 | \( 1 + 4.96iT - 53T^{2} \) |
| 59 | \( 1 - 14.9T + 59T^{2} \) |
| 61 | \( 1 + 5.71T + 61T^{2} \) |
| 67 | \( 1 - 2.49iT - 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 + 6.57iT - 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
| 97 | \( 1 - 7.26iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.23359350780052784962812480444, −8.939335188052240161829159984351, −8.029175015406453860359937270024, −7.40650367220298492048678036230, −6.70331922865792891912973503441, −6.43790540208841160615471229342, −5.45716213737517118225381230816, −3.85737884265958049786410053146, −2.39915871121573909540101295442, −1.75178711593464028137921263172,
0.67969249760368036503610383658, 2.47918037507947566295807095075, 3.36216070336536354018401584333, 4.31442452913386389676619475639, 5.17207350101692397880256264040, 5.85825743783092801340174717833, 7.37319655046602089194000311580, 8.395068584329088745335062321151, 9.229108905440709995855537706271, 9.839452418611490936488128618242