| L(s) = 1 | − 5-s + 2·7-s + 13-s + 5·17-s − 6·19-s + 2·23-s − 4·25-s − 9·29-s + 2·31-s − 2·35-s − 3·37-s + 5·41-s + 2·47-s − 3·49-s − 9·53-s + 8·59-s + 6·61-s − 65-s − 2·67-s + 12·71-s − 2·73-s + 10·79-s + 6·83-s − 5·85-s + 9·89-s + 2·91-s + 6·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 0.277·13-s + 1.21·17-s − 1.37·19-s + 0.417·23-s − 4/5·25-s − 1.67·29-s + 0.359·31-s − 0.338·35-s − 0.493·37-s + 0.780·41-s + 0.291·47-s − 3/7·49-s − 1.23·53-s + 1.04·59-s + 0.768·61-s − 0.124·65-s − 0.244·67-s + 1.42·71-s − 0.234·73-s + 1.12·79-s + 0.658·83-s − 0.542·85-s + 0.953·89-s + 0.209·91-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923912293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923912293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.81636496217109, −15.20298062004337, −14.74160510301610, −14.37831711373525, −13.65627848201924, −13.05482803579212, −12.47866917102462, −12.00226239159769, −11.14559089380602, −11.11280004509611, −10.29269492256047, −9.621771882723607, −9.050772648013521, −8.223240187686986, −7.982924513765691, −7.342846013723441, −6.610636820342661, −5.868421328425316, −5.308193101184314, −4.580826450508769, −3.876195791908593, −3.374824236504339, −2.275268123684058, −1.639651334855997, −0.5931657359043535,
0.5931657359043535, 1.639651334855997, 2.275268123684058, 3.374824236504339, 3.876195791908593, 4.580826450508769, 5.308193101184314, 5.868421328425316, 6.610636820342661, 7.342846013723441, 7.982924513765691, 8.223240187686986, 9.050772648013521, 9.621771882723607, 10.29269492256047, 11.11280004509611, 11.14559089380602, 12.00226239159769, 12.47866917102462, 13.05482803579212, 13.65627848201924, 14.37831711373525, 14.74160510301610, 15.20298062004337, 15.81636496217109
