| L(s) = 1 | + 0.598i·2-s + 1.31i·3-s + 1.64·4-s + (1.14 − 1.91i)5-s − 0.786·6-s + 0.976i·7-s + 2.17i·8-s + 1.27·9-s + (1.14 + 0.686i)10-s + 11-s + 2.15i·12-s − 2.37i·13-s − 0.584·14-s + (2.52 + 1.50i)15-s + 1.98·16-s + 2.72i·17-s + ⋯ |
| L(s) = 1 | + 0.423i·2-s + 0.759i·3-s + 0.821·4-s + (0.512 − 0.858i)5-s − 0.321·6-s + 0.369i·7-s + 0.770i·8-s + 0.423·9-s + (0.363 + 0.216i)10-s + 0.301·11-s + 0.623i·12-s − 0.657i·13-s − 0.156·14-s + (0.651 + 0.389i)15-s + 0.495·16-s + 0.661i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.360322270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360322270\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.14 + 1.91i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.598iT - 2T^{2} \) |
| 3 | \( 1 - 1.31iT - 3T^{2} \) |
| 7 | \( 1 - 0.976iT - 7T^{2} \) |
| 13 | \( 1 + 2.37iT - 13T^{2} \) |
| 17 | \( 1 - 2.72iT - 17T^{2} \) |
| 23 | \( 1 + 2.63iT - 23T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 + 3.93T + 31T^{2} \) |
| 37 | \( 1 + 5.08iT - 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 - 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 9.84iT - 47T^{2} \) |
| 53 | \( 1 - 0.422iT - 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 - 6.89T + 61T^{2} \) |
| 67 | \( 1 - 2.42iT - 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.27iT - 73T^{2} \) |
| 79 | \( 1 - 0.722T + 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 12.9iT - 97T^{2} \) |
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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
−10.15674915568357698814083777939, −9.158353130152508117701002480091, −8.481590373630523470510337739047, −7.60741055648343228649904209461, −6.51119749478637790608700249619, −5.78016446073483136088366316565, −4.97817607873448290352153011809, −4.02395685173128213326570679434, −2.67108719381556544068713702742, −1.44898032497550440766581836533,
1.29040252269434206020507363731, 2.18523337420772870862684181561, 3.15264689941423239770314564812, 4.30058720577254909702317912881, 5.83242493521251682766077909390, 6.67216851111952203223031911934, 7.07877452351836602173647729316, 7.76683677093262782348223086882, 9.174786589347113679748903001544, 9.993366606509861410462498120698
