L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.473 − 0.397i)5-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.580 + 0.211i)10-s + (−0.280 + 1.59i)13-s + (0.173 + 0.984i)16-s + (1.52 + 0.553i)17-s + 18-s + 0.618·20-s + (−0.107 + 0.608i)25-s + (0.809 − 1.40i)26-s + (1.52 − 0.553i)29-s + (0.173 − 0.984i)32-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.473 − 0.397i)5-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.580 + 0.211i)10-s + (−0.280 + 1.59i)13-s + (0.173 + 0.984i)16-s + (1.52 + 0.553i)17-s + 18-s + 0.618·20-s + (−0.107 + 0.608i)25-s + (0.809 − 1.40i)26-s + (1.52 − 0.553i)29-s + (0.173 − 0.984i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7382201982\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7382201982\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \) |
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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
−9.682768973532610147347256220755, −9.039912127445472341223635810122, −8.293101623794104545501494889382, −7.61203695714711551966189002497, −6.54998974306443450148532894833, −5.83944118608165071152378528617, −4.70985171622714070091941623491, −3.47346839563428514799461255852, −2.40903223547916532589738432403, −1.37806062641709026246631063626,
0.912850641910967865228431253617, 2.62435094381009984568625753511, 3.15829891789116712813898812895, 5.07778629244403469487860978499, 5.78461283912455859771564388057, 6.39892520580659288600134741642, 7.49652950146497905819107035798, 8.058929506133913425033506855688, 8.849577931142348211055459580356, 9.753068678713352110389568489450