L(s) = 1 | + (−1.53 + 1.11i)2-s + (0.809 − 2.48i)4-s + (−0.309 + 0.951i)7-s + (0.951 + 2.92i)8-s + (0.587 − 0.809i)11-s + (−0.587 − 1.80i)14-s + (−2.61 − 1.90i)16-s + 1.90i·22-s + 1.90·23-s + (0.309 + 0.951i)25-s + (2.11 + 1.53i)28-s + (−0.363 + 1.11i)29-s + 3.07·32-s + (−0.190 + 0.587i)37-s + 1.61·43-s + (−1.53 − 2.11i)44-s + ⋯ |
L(s) = 1 | + (−1.53 + 1.11i)2-s + (0.809 − 2.48i)4-s + (−0.309 + 0.951i)7-s + (0.951 + 2.92i)8-s + (0.587 − 0.809i)11-s + (−0.587 − 1.80i)14-s + (−2.61 − 1.90i)16-s + 1.90i·22-s + 1.90·23-s + (0.309 + 0.951i)25-s + (2.11 + 1.53i)28-s + (−0.363 + 1.11i)29-s + 3.07·32-s + (−0.190 + 0.587i)37-s + 1.61·43-s + (−1.53 − 2.11i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0237 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0237 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4603585022\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4603585022\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-0.587 + 0.809i)T \) |
good | 2 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - 1.90T + T^{2} \) |
| 29 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61T + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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
−10.79376272061957872363800837253, −9.488302408199264340459984915077, −9.072958959368250335261617002486, −8.514104180368091805562152184800, −7.42602950802704309884997689260, −6.67586411438720131305083836267, −5.85018459067842607722955738099, −5.08214499233815258295364921669, −3.00403653044493241788977459216, −1.34739399174941938019478509485,
1.00183530503990345042120480982, 2.37118010573218337153350410883, 3.54609656601520768809208747270, 4.50254563103778802406453216618, 6.55842952760049731892465417006, 7.28409487718031471102412253892, 8.006120843950209007586739503307, 9.156423215927911995231321270030, 9.534521051531393810509189631233, 10.50020098619143192761601827958