L(s) = 1 | + (1.16 − 1.59i)2-s + (−0.951 + 0.309i)3-s + (−0.896 − 2.76i)4-s + (0.891 − 0.453i)5-s + (−0.610 + 1.87i)6-s + (−3.57 − 1.16i)8-s + (0.809 − 0.587i)9-s + (0.309 − 1.95i)10-s + (−0.734 − 0.533i)11-s + (1.70 + 2.34i)12-s + (0.587 + 0.809i)13-s + (−0.707 + 0.707i)15-s + (−3.65 + 2.65i)16-s − 1.97i·18-s + (−2.05 − 2.05i)20-s + ⋯ |
L(s) = 1 | + (1.16 − 1.59i)2-s + (−0.951 + 0.309i)3-s + (−0.896 − 2.76i)4-s + (0.891 − 0.453i)5-s + (−0.610 + 1.87i)6-s + (−3.57 − 1.16i)8-s + (0.809 − 0.587i)9-s + (0.309 − 1.95i)10-s + (−0.734 − 0.533i)11-s + (1.70 + 2.34i)12-s + (0.587 + 0.809i)13-s + (−0.707 + 0.707i)15-s + (−3.65 + 2.65i)16-s − 1.97i·18-s + (−2.05 − 2.05i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.368961702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368961702\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (-0.891 + 0.453i)T \) |
| 13 | \( 1 + (-0.587 - 0.809i)T \) |
good | 2 | \( 1 + (-1.16 + 1.59i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 1.17iT - T^{2} \) |
| 47 | \( 1 + (-0.297 + 0.0966i)T + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.87 - 0.610i)T + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
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
−10.08831755130948594678206772602, −9.558604571349753043148508234650, −8.677839925449933497985150915828, −6.59121833337239780133057247212, −5.92025745519899664545955788793, −5.22850951563441828336108738505, −4.53031245259855072937535513369, −3.55054489984428286056153619092, −2.25822554770458533678005329018, −1.10713423397349020545853462097,
2.52404999577901244857108114913, 3.82242670165022854187933921397, 5.05381044362367564822996725311, 5.48028018714523120290135158470, 6.24496009572841904031230285485, 6.93029784134682791485207693416, 7.63586578555980103540946805384, 8.500824382703307281148769698521, 9.730871595444087969595038185905, 10.68389592298953473301666234419