L(s) = 1 | + (1.35 − 2.34i)5-s + (10.0 − 5.79i)7-s + (−8.54 + 4.93i)11-s + (−0.296 + 0.513i)13-s + 8.87·17-s + 14.0i·19-s + (18.2 + 10.5i)23-s + (8.82 + 15.2i)25-s + (10.1 + 17.6i)29-s + (14.3 + 8.27i)31-s − 31.4i·35-s + 40.6·37-s + (−21.2 + 36.7i)41-s + (32.2 − 18.6i)43-s + (−1.57 + 0.907i)47-s + ⋯ |
L(s) = 1 | + (0.271 − 0.469i)5-s + (1.43 − 0.828i)7-s + (−0.777 + 0.448i)11-s + (−0.0227 + 0.0394i)13-s + 0.522·17-s + 0.742i·19-s + (0.794 + 0.458i)23-s + (0.352 + 0.611i)25-s + (0.350 + 0.607i)29-s + (0.462 + 0.266i)31-s − 0.898i·35-s + 1.09·37-s + (−0.517 + 0.896i)41-s + (0.749 − 0.432i)43-s + (−0.0334 + 0.0193i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.631529187\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.631529187\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.35 + 2.34i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-10.0 + 5.79i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (8.54 - 4.93i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (0.296 - 0.513i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 8.87T + 289T^{2} \) |
| 19 | \( 1 - 14.0iT - 361T^{2} \) |
| 23 | \( 1 + (-18.2 - 10.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-10.1 - 17.6i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-14.3 - 8.27i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (21.2 - 36.7i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-32.2 + 18.6i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.57 - 0.907i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 21.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-76.6 - 44.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (36.4 + 63.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-38.3 - 22.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 111. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + (8.30 - 4.79i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-73.6 + 42.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (3.59 + 6.22i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.074263346815617258869692093662, −8.142420731740603171097588745385, −7.69644407616341194290406935043, −6.90387502919823058756082839411, −5.62523953355446421280692390051, −4.97243286081163299428051818956, −4.34506955598628946725427806847, −3.11868716455389593304523496984, −1.76911424712179096315048078091, −1.00229021932008264780434647622,
0.898317860190383048832897532570, 2.31899506577021257293228242741, 2.83017989587920964661908938777, 4.34535740741312771331920945735, 5.14157595650859808454279149722, 5.79468124404246987683695779937, 6.75463250706584008337662850208, 7.76360681417203824303537150592, 8.319193106212845557599203166157, 9.011389513810267994103581993980