L(s) = 1 | + 2-s + 4-s + 5-s + 2.79·7-s + 8-s + 10-s − 3.79·11-s + 1.20·13-s + 2.79·14-s + 16-s + 3.79·17-s + 1.20·19-s + 20-s − 3.79·22-s − 23-s + 25-s + 1.20·26-s + 2.79·28-s + 1.58·29-s + 10.3·31-s + 32-s + 3.79·34-s + 2.79·35-s − 4·37-s + 1.20·38-s + 40-s + 2.20·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.05·7-s + 0.353·8-s + 0.316·10-s − 1.14·11-s + 0.335·13-s + 0.746·14-s + 0.250·16-s + 0.919·17-s + 0.277·19-s + 0.223·20-s − 0.808·22-s − 0.208·23-s + 0.200·25-s + 0.237·26-s + 0.527·28-s + 0.293·29-s + 1.86·31-s + 0.176·32-s + 0.650·34-s + 0.471·35-s − 0.657·37-s + 0.196·38-s + 0.158·40-s + 0.344·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.438551882\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.438551882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 - 2.79T + 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 - 3.79T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 29 | \( 1 - 1.58T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 7.16T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 3.37T + 61T^{2} \) |
| 67 | \( 1 + 7.16T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 3.16T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.051919674050273878786995763399, −8.071131541486010082244697026693, −7.71473028010385140850939261838, −6.62375420475338922949842781516, −5.72995739153189819484233326584, −5.12655129536357619195484401242, −4.42748173398813786241702250848, −3.22514659993522307751921344136, −2.34739889695145047369938826561, −1.22508210605556772251166314152,
1.22508210605556772251166314152, 2.34739889695145047369938826561, 3.22514659993522307751921344136, 4.42748173398813786241702250848, 5.12655129536357619195484401242, 5.72995739153189819484233326584, 6.62375420475338922949842781516, 7.71473028010385140850939261838, 8.071131541486010082244697026693, 9.051919674050273878786995763399