L(s) = 1 | + (0.755 + 0.654i)2-s + (−0.281 + 0.959i)3-s + (0.142 + 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.841 + 0.540i)6-s + (0.540 + 1.84i)7-s + (−0.540 + 0.841i)8-s + (−0.841 − 0.540i)9-s + (0.281 − 0.959i)10-s + (−0.989 − 0.142i)12-s + (−0.797 + 1.74i)14-s + (0.989 − 0.142i)15-s + (−0.959 + 0.281i)16-s + (−0.281 − 0.959i)18-s + (0.841 − 0.540i)20-s − 1.91·21-s + ⋯ |
L(s) = 1 | + (0.755 + 0.654i)2-s + (−0.281 + 0.959i)3-s + (0.142 + 0.989i)4-s + (−0.415 − 0.909i)5-s + (−0.841 + 0.540i)6-s + (0.540 + 1.84i)7-s + (−0.540 + 0.841i)8-s + (−0.841 − 0.540i)9-s + (0.281 − 0.959i)10-s + (−0.989 − 0.142i)12-s + (−0.797 + 1.74i)14-s + (0.989 − 0.142i)15-s + (−0.959 + 0.281i)16-s + (−0.281 − 0.959i)18-s + (0.841 − 0.540i)20-s − 1.91·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.312732682\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.312732682\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.755 - 0.654i)T \) |
| 3 | \( 1 + (0.281 - 0.959i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.755 + 0.654i)T \) |
good | 7 | \( 1 + (-0.540 - 1.84i)T + (-0.841 + 0.540i)T^{2} \) |
| 11 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 13 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 17 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 19 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 29 | \( 1 + (-1.80 - 0.258i)T + (0.959 + 0.281i)T^{2} \) |
| 31 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 37 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 41 | \( 1 + (0.512 - 0.234i)T + (0.654 - 0.755i)T^{2} \) |
| 43 | \( 1 + (0.153 + 0.239i)T + (-0.415 + 0.909i)T^{2} \) |
| 47 | \( 1 - 0.830iT - T^{2} \) |
| 53 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 59 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 61 | \( 1 + (-1.07 + 1.66i)T + (-0.415 - 0.909i)T^{2} \) |
| 67 | \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 71 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.449 + 0.983i)T + (-0.654 - 0.755i)T^{2} \) |
| 89 | \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
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
−9.956142675494984270075029906284, −8.965438192340205125459960883201, −8.534943360122828773677947974381, −7.990699117932095726187712717020, −6.45599115554329977559440263545, −5.77001370672906382040129791376, −4.97596964290470858991330186164, −4.60838064467782582915054952523, −3.43272717323129941187939444467, −2.34382029550534758058044397445,
0.915473360454769807540056941309, 2.14713285572932258158731891081, 3.34507296239879856674653387318, 4.14426744075094985805563795628, 5.10735297524540209165950209719, 6.31825644664479326940500639772, 6.87886779976022445812265299663, 7.55510509184160229573875117016, 8.339453798843349127985531090260, 9.998058628865432692533740287266