L(s) = 1 | + 3i·3-s + 32i·7-s − 9·9-s − 36·11-s − 10i·13-s + 78i·17-s + 140·19-s − 96·21-s + 192i·23-s − 27i·27-s − 6·29-s + 16·31-s − 108i·33-s + 34i·37-s + 30·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.72i·7-s − 0.333·9-s − 0.986·11-s − 0.213i·13-s + 1.11i·17-s + 1.69·19-s − 0.997·21-s + 1.74i·23-s − 0.192i·27-s − 0.0384·29-s + 0.0926·31-s − 0.569i·33-s + 0.151i·37-s + 0.123·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.119886271\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.119886271\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 32iT - 343T^{2} \) |
| 11 | \( 1 + 36T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 78iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 140T + 6.85e3T^{2} \) |
| 23 | \( 1 - 192iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 390T + 6.89e4T^{2} \) |
| 43 | \( 1 - 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 408iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 114iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 516T + 2.05e5T^{2} \) |
| 61 | \( 1 + 58T + 2.26e5T^{2} \) |
| 67 | \( 1 + 892iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 120T + 3.57e5T^{2} \) |
| 73 | \( 1 + 646iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 732iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 194iT - 9.12e5T^{2} \) |
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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
−9.704586365259199307735183401707, −9.154603467234969014803669795512, −8.247308025664283323013074585573, −7.63726016295088048933681421097, −6.23502171738817250648913039889, −5.42606002259746514607998456591, −5.08432990551913592494471533878, −3.53293852488974195148669830579, −2.82331758377686174962522834721, −1.65292590279535744207053773131,
0.29087195048078879486558595838, 1.06757517731194835601524376552, 2.52110175476443598819491586999, 3.51887377046012711271589194477, 4.64368969647486511402003837222, 5.41469813221402347453268415132, 6.77177939909857387783914187674, 7.19341432647054811395914329457, 7.87458782606368485900950298732, 8.776823205098561513861928352368