L(s) = 1 | + (−0.974 − 0.222i)3-s + (0.222 + 0.974i)4-s + (0.360 − 1.57i)7-s + (0.900 + 0.433i)9-s − i·12-s + (0.556 − 0.268i)13-s + (−0.900 + 0.433i)16-s + (0.602 − 0.137i)19-s + (−0.702 + 1.45i)21-s + (−0.222 − 0.974i)25-s + (−0.781 − 0.623i)27-s + 1.61·28-s + (−1.26 − 1.00i)31-s + (−0.222 + 0.974i)36-s + (−0.268 + 0.556i)37-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.222i)3-s + (0.222 + 0.974i)4-s + (0.360 − 1.57i)7-s + (0.900 + 0.433i)9-s − i·12-s + (0.556 − 0.268i)13-s + (−0.900 + 0.433i)16-s + (0.602 − 0.137i)19-s + (−0.702 + 1.45i)21-s + (−0.222 − 0.974i)25-s + (−0.781 − 0.623i)27-s + 1.61·28-s + (−1.26 − 1.00i)31-s + (−0.222 + 0.974i)36-s + (−0.268 + 0.556i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9479089110\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9479089110\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.974 + 0.222i)T \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 5 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (-0.360 + 1.57i)T + (-0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-0.556 + 0.268i)T + (0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.602 + 0.137i)T + (0.900 - 0.433i)T^{2} \) |
| 23 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (1.26 + 1.00i)T + (0.222 + 0.974i)T^{2} \) |
| 37 | \( 1 + (0.268 - 0.556i)T + (-0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1.26 + 1.00i)T + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-1.57 - 0.360i)T + (0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.483 + 0.385i)T + (0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2} \) |
| 83 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 + (-1.57 + 0.360i)T + (0.900 - 0.433i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.899165881763060334203911383374, −7.88400497987356771135902019543, −7.48672024357924725841629784740, −6.85740551826042833432315854811, −6.04860350983906191130208908789, −5.00650714893477179702905186032, −4.11333105819360385564572359716, −3.62442701967029038241153605060, −2.13278273125476155769238280725, −0.800422261972713066664649704950,
1.31016054440299041107434375314, 2.21223559519177904096738983261, 3.58886033954051034692267720033, 4.82799088096735540646352264618, 5.45512431409651125413961067415, 5.82828481389231086585531353608, 6.63440995374880157456025891907, 7.46787430133799321981060530066, 8.677429515708931493482146371701, 9.317256861073392824076246013254