L(s) = 1 | + (0.433 + 0.900i)3-s + (−0.974 − 0.222i)4-s + (0.623 + 0.781i)7-s + (−0.623 + 0.781i)9-s + (−0.222 − 0.974i)12-s + (−0.433 + 0.900i)13-s + (0.900 + 0.433i)16-s + (0.158 − 0.158i)19-s + (−0.433 + 0.900i)21-s + (−0.781 − 0.623i)25-s + (−0.974 − 0.222i)27-s + (−0.433 − 0.900i)28-s + (−0.752 + 0.752i)31-s + (0.781 − 0.623i)36-s + (0.900 + 0.566i)37-s + ⋯ |
L(s) = 1 | + (0.433 + 0.900i)3-s + (−0.974 − 0.222i)4-s + (0.623 + 0.781i)7-s + (−0.623 + 0.781i)9-s + (−0.222 − 0.974i)12-s + (−0.433 + 0.900i)13-s + (0.900 + 0.433i)16-s + (0.158 − 0.158i)19-s + (−0.433 + 0.900i)21-s + (−0.781 − 0.623i)25-s + (−0.974 − 0.222i)27-s + (−0.433 − 0.900i)28-s + (−0.752 + 0.752i)31-s + (0.781 − 0.623i)36-s + (0.900 + 0.566i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9468746799\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9468746799\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.433 - 0.900i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
| 13 | \( 1 + (0.433 - 0.900i)T \) |
good | 2 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 5 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 11 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 + (-0.158 + 0.158i)T - iT^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (0.752 - 0.752i)T - iT^{2} \) |
| 37 | \( 1 + (-0.900 - 0.566i)T + (0.433 + 0.900i)T^{2} \) |
| 41 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 43 | \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \) |
| 47 | \( 1 + (0.974 + 0.222i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (0.781 - 0.623i)T^{2} \) |
| 61 | \( 1 + (-0.433 + 0.0990i)T + (0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 + (-0.467 - 0.467i)T + iT^{2} \) |
| 71 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 73 | \( 1 + (1.68 - 0.189i)T + (0.974 - 0.222i)T^{2} \) |
| 79 | \( 1 - 1.94T + T^{2} \) |
| 83 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 89 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 97 | \( 1 + (-1.40 + 1.40i)T - iT^{2} \) |
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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
−9.586619333586390535356396313429, −8.937208489676221011825933301633, −8.364504675536949194924763014380, −7.63672005738963936373469978490, −6.25890188398263959222620247625, −5.35988230972950969854993162999, −4.72036808833224065283092882147, −4.10245481618174907417991579011, −2.98011391789669407422798903893, −1.82547657097284699066770897401,
0.68839286266825953321751748238, 2.01271906506548941092947376305, 3.33169339314016129837713214839, 4.03050141757219275924555093360, 5.14062822804627522127128624531, 5.85326287786226287313445575242, 7.11105712030937989984041120487, 7.72516090422048761251264961294, 8.142109070419417190091924976784, 9.045621834957695335375223947233