L(s) = 1 | + (−0.400 + 0.193i)5-s + (0.222 − 0.974i)9-s + (−0.277 − 1.21i)13-s − 0.867i·17-s + (−0.499 + 0.626i)25-s + (−0.222 − 0.974i)29-s + (1.52 + 0.347i)37-s − 1.56i·41-s + (0.0990 + 0.433i)45-s + (−0.222 + 0.974i)49-s + (1.62 − 0.781i)53-s + (0.678 − 0.541i)61-s + (0.346 + 0.433i)65-s + (−0.376 + 0.781i)73-s + (−0.900 − 0.433i)81-s + ⋯ |
L(s) = 1 | + (−0.400 + 0.193i)5-s + (0.222 − 0.974i)9-s + (−0.277 − 1.21i)13-s − 0.867i·17-s + (−0.499 + 0.626i)25-s + (−0.222 − 0.974i)29-s + (1.52 + 0.347i)37-s − 1.56i·41-s + (0.0990 + 0.433i)45-s + (−0.222 + 0.974i)49-s + (1.62 − 0.781i)53-s + (0.678 − 0.541i)61-s + (0.346 + 0.433i)65-s + (−0.376 + 0.781i)73-s + (−0.900 − 0.433i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9759781007\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9759781007\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (0.222 + 0.974i)T \) |
good | 3 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 5 | \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 11 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 13 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + 0.867iT - T^{2} \) |
| 19 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 23 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 37 | \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + 1.56iT - T^{2} \) |
| 43 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 53 | \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.678 + 0.541i)T + (0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 71 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 73 | \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \) |
| 79 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.846 + 1.75i)T + (-0.623 + 0.781i)T^{2} \) |
| 97 | \( 1 + (-1.22 - 0.974i)T + (0.222 + 0.974i)T^{2} \) |
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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
−9.436653452355185979337249571950, −8.464180545057106500672646560095, −7.63152267431950739741544709589, −7.07487082890963847756262975541, −6.06145940840212701042762342774, −5.33410026124228623930235987615, −4.21138659531628813221436664589, −3.42068950481351453116891051563, −2.47320129962803857641586932910, −0.75785906488049320529726083398,
1.58754385144782791207648641768, 2.59581044583718816769976398635, 3.98998805816878427026662599927, 4.51451505016131034534042735505, 5.49842263806496142024307812322, 6.46242044458516705699092583169, 7.28429392937516010969451124056, 8.032275090676452799607818280638, 8.680690917630154241629325109630, 9.586656604318756660675659741351