L(s) = 1 | + 1.41·2-s − 2.64·5-s − 7-s − 2.82·8-s − 3.74·10-s − 1.74·13-s − 1.41·14-s − 4.00·16-s + 0.182·17-s + 1.74·19-s + 6.70·23-s + 2.00·25-s − 2.46·26-s − 6.70·29-s − 9.48·31-s + 0.258·34-s + 2.64·35-s − 11.4·37-s + 2.46·38-s + 7.48·40-s − 5.65·41-s + 43-s + 9.48·46-s − 0.182·47-s + 49-s + ⋯ |
L(s) = 1 | + 1.00·2-s − 1.18·5-s − 0.377·7-s − 0.999·8-s − 1.18·10-s − 0.483·13-s − 0.377·14-s − 1.00·16-s + 0.0443·17-s + 0.399·19-s + 1.39·23-s + 0.400·25-s − 0.483·26-s − 1.24·29-s − 1.70·31-s + 0.0443·34-s + 0.447·35-s − 1.88·37-s + 0.399·38-s + 1.18·40-s − 0.883·41-s + 0.152·43-s + 1.39·46-s − 0.0266·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.116302560\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.116302560\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.182T + 17T^{2} \) |
| 19 | \( 1 - 1.74T + 19T^{2} \) |
| 23 | \( 1 - 6.70T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 9.48T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 0.182T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 - 3.01T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 1.41T + 71T^{2} \) |
| 73 | \( 1 - 8.25T + 73T^{2} \) |
| 79 | \( 1 + 1.48T + 79T^{2} \) |
| 83 | \( 1 - 5.83T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.68007538373333361161680149321, −7.12220518844282945236988913993, −6.51694658138679680755276678497, −5.32686132755265527637734014207, −5.21296378410675720868445627944, −4.22565761302153338441261506745, −3.47056802819728049973090436880, −3.28662973785159314069378584008, −2.00542259431317959656581270429, −0.43367795918564847594465123156,
0.43367795918564847594465123156, 2.00542259431317959656581270429, 3.28662973785159314069378584008, 3.47056802819728049973090436880, 4.22565761302153338441261506745, 5.21296378410675720868445627944, 5.32686132755265527637734014207, 6.51694658138679680755276678497, 7.12220518844282945236988913993, 7.68007538373333361161680149321