L(s) = 1 | + (0.723 − 0.690i)3-s + (0.580 − 0.814i)4-s + (0.841 − 0.540i)7-s + (0.0475 − 0.998i)9-s + (−0.142 − 0.989i)12-s + (−1.32 + 1.04i)13-s + (−0.327 − 0.945i)16-s + (0.0688 + 1.44i)19-s + (0.235 − 0.971i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.0475 − 0.998i)28-s + (−1.21 + 0.486i)31-s + (−0.786 − 0.618i)36-s + (−0.928 + 1.60i)37-s + ⋯ |
L(s) = 1 | + (0.723 − 0.690i)3-s + (0.580 − 0.814i)4-s + (0.841 − 0.540i)7-s + (0.0475 − 0.998i)9-s + (−0.142 − 0.989i)12-s + (−1.32 + 1.04i)13-s + (−0.327 − 0.945i)16-s + (0.0688 + 1.44i)19-s + (0.235 − 0.971i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.0475 − 0.998i)28-s + (−1.21 + 0.486i)31-s + (−0.786 − 0.618i)36-s + (−0.928 + 1.60i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.602737434\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.602737434\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.723 + 0.690i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
good | 2 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 5 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 11 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 13 | \( 1 + (1.32 - 1.04i)T + (0.235 - 0.971i)T^{2} \) |
| 17 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.0688 - 1.44i)T + (-0.995 + 0.0950i)T^{2} \) |
| 23 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (1.21 - 0.486i)T + (0.723 - 0.690i)T^{2} \) |
| 37 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 43 | \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 53 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 59 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 61 | \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \) |
| 71 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 73 | \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (-0.928 - 0.371i)T^{2} \) |
| 89 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 97 | \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.641608585153903639608320793783, −8.736465614037908357495597130231, −7.82873753341963400956230512039, −7.11837346862947132691791629691, −6.62792026717703991879626228436, −5.44069915071064958697467493350, −4.57444909243038960959960517855, −3.32853980859802339555987946715, −2.05583911572033653276860938764, −1.44652488575096348734718058420,
2.24994484112791713038730641032, 2.71934854536296136627825984997, 3.82437883834550200292180062801, 4.86816361716712214835678509527, 5.49306636568383567415217093069, 7.08979048464832999857493964424, 7.51951153204532645304711722703, 8.426343022684034676252611809218, 8.915953645358095207101127598449, 9.838511727462246247589493675591