L(s) = 1 | + 2·5-s − 2·11-s − 2·17-s − 2·23-s − 25-s + 6·29-s − 4·31-s − 6·37-s + 2·41-s − 6·53-s − 4·55-s + 12·59-s − 12·61-s + 12·67-s + 10·71-s + 12·73-s + 12·79-s + 12·83-s − 4·85-s − 14·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.603·11-s − 0.485·17-s − 0.417·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.986·37-s + 0.312·41-s − 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.53·61-s + 1.46·67-s + 1.18·71-s + 1.40·73-s + 1.35·79-s + 1.31·83-s − 0.433·85-s − 1.48·89-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.49728545016949, −15.02534207152513, −14.14762733862665, −13.95645897561193, −13.47176120914847, −12.75953593680346, −12.45603529036716, −11.75615314246829, −11.03549367423849, −10.64935449843986, −10.02956132839025, −9.564411167676348, −9.046886513437834, −8.297569709882017, −7.912937612969561, −7.083945070287202, −6.518688622454353, −6.023463160281178, −5.248158910299300, −4.952074548269805, −4.004183412366822, −3.371195476963263, −2.411091682289289, −2.077289354049083, −1.102439462141840, 0,
1.102439462141840, 2.077289354049083, 2.411091682289289, 3.371195476963263, 4.004183412366822, 4.952074548269805, 5.248158910299300, 6.023463160281178, 6.518688622454353, 7.083945070287202, 7.912937612969561, 8.297569709882017, 9.046886513437834, 9.564411167676348, 10.02956132839025, 10.64935449843986, 11.03549367423849, 11.75615314246829, 12.45603529036716, 12.75953593680346, 13.47176120914847, 13.95645897561193, 14.14762733862665, 15.02534207152513, 15.49728545016949