L(s) = 1 | + (−0.988 + 0.149i)3-s + (−0.733 + 0.680i)7-s + (0.955 − 0.294i)9-s + (0.440 + 1.92i)13-s + (0.733 + 1.26i)19-s + (0.623 − 0.781i)21-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.0747 + 0.129i)31-s + (0.123 + 1.64i)37-s + (−0.722 − 1.84i)39-s + (1.19 − 1.49i)43-s + (0.0747 − 0.997i)49-s + (−0.914 − 1.14i)57-s + (−0.134 − 1.79i)61-s + ⋯ |
L(s) = 1 | + (−0.988 + 0.149i)3-s + (−0.733 + 0.680i)7-s + (0.955 − 0.294i)9-s + (0.440 + 1.92i)13-s + (0.733 + 1.26i)19-s + (0.623 − 0.781i)21-s + (−0.733 − 0.680i)25-s + (−0.900 + 0.433i)27-s + (−0.0747 + 0.129i)31-s + (0.123 + 1.64i)37-s + (−0.722 − 1.84i)39-s + (1.19 − 1.49i)43-s + (0.0747 − 0.997i)49-s + (−0.914 − 1.14i)57-s + (−0.134 − 1.79i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6039969768\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6039969768\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.988 - 0.149i)T \) |
| 7 | \( 1 + (0.733 - 0.680i)T \) |
good | 5 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (-0.440 - 1.92i)T + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.0747 - 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.123 - 1.64i)T + (-0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (-1.19 + 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \) |
| 79 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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
−11.23882470857432918760872504502, −10.10873822640107706742314434413, −9.542405071276280526906127657123, −8.611254634631003346435018234611, −7.27387488018695372656256939126, −6.35843250444775185597883629755, −5.80893858356009987357686940213, −4.58053483721138891425619758812, −3.58989539608701462091752793780, −1.81313514156178756525041020053,
0.850387021171205720859953640240, 2.97447454796092782650989884802, 4.17060743136478883305388648157, 5.42272541426778125314327821034, 6.04364205242385466211708240671, 7.21533881119041739788078534254, 7.72971333118621795649014534876, 9.201820897671545335986412149331, 10.06400492228243681979513855961, 10.78633089672602809975906214180