L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.613 + 1.64i)3-s + (0.415 + 0.909i)4-s + (0.373 − 1.71i)6-s + (0.142 − 0.989i)8-s + (−1.57 + 1.36i)9-s + (0.281 + 1.95i)11-s + (−1.24 + 1.24i)12-s + (−0.654 + 0.755i)16-s + (−0.909 − 1.41i)17-s + (2.06 − 0.296i)18-s + (0.0855 − 1.19i)19-s + (0.822 − 1.80i)22-s + (1.71 − 0.373i)24-s + (0.959 + 0.281i)25-s + ⋯ |
L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.613 + 1.64i)3-s + (0.415 + 0.909i)4-s + (0.373 − 1.71i)6-s + (0.142 − 0.989i)8-s + (−1.57 + 1.36i)9-s + (0.281 + 1.95i)11-s + (−1.24 + 1.24i)12-s + (−0.654 + 0.755i)16-s + (−0.909 − 1.41i)17-s + (2.06 − 0.296i)18-s + (0.0855 − 1.19i)19-s + (0.822 − 1.80i)22-s + (1.71 − 0.373i)24-s + (0.959 + 0.281i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00217 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00217 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7515851560\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7515851560\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 89 | \( 1 + (-0.959 + 0.281i)T \) |
good | 3 | \( 1 + (-0.613 - 1.64i)T + (-0.755 + 0.654i)T^{2} \) |
| 5 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 7 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 11 | \( 1 + (-0.281 - 1.95i)T + (-0.959 + 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 17 | \( 1 + (0.909 + 1.41i)T + (-0.415 + 0.909i)T^{2} \) |
| 19 | \( 1 + (-0.0855 + 1.19i)T + (-0.989 - 0.142i)T^{2} \) |
| 23 | \( 1 + (-0.989 - 0.142i)T^{2} \) |
| 29 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 31 | \( 1 + (-0.989 + 0.142i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (1.32 + 0.494i)T + (0.755 + 0.654i)T^{2} \) |
| 43 | \( 1 + (-1.56 - 1.17i)T + (0.281 + 0.959i)T^{2} \) |
| 47 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 53 | \( 1 + (-0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (0.148 - 0.398i)T + (-0.755 - 0.654i)T^{2} \) |
| 61 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 67 | \( 1 + (-0.627 + 1.37i)T + (-0.654 - 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 73 | \( 1 + (-0.368 + 0.425i)T + (-0.142 - 0.989i)T^{2} \) |
| 79 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.148 + 0.682i)T + (-0.909 - 0.415i)T^{2} \) |
| 97 | \( 1 + (-0.186 + 1.29i)T + (-0.959 - 0.281i)T^{2} \) |
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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
−10.66886667392446840258824536712, −9.746145499182694545121292796332, −9.330251285304499287017071035740, −8.824183198511035470665139816233, −7.56823431775412822501222110583, −6.82115131374994817406844522708, −4.78798972067198893608645124043, −4.48278786803441124014085460668, −3.14113533431746785323672389963, −2.27520708070380661986929873867,
1.06365130455925400985305775541, 2.20654960844176425300557546976, 3.52858218459818191099653005821, 5.66323055052109113858823318291, 6.30858795323586015705100497639, 6.93865000781591366778755451926, 8.125941891652855746568624004173, 8.387989630863804483044316002461, 9.026909447146031571690579567990, 10.42590941626356653312882124959