| L(s) = 1 | + 1.51·3-s + 5-s + 0.103·7-s − 0.696·9-s + 3.13·11-s − 1.31·13-s + 1.51·15-s − 2.54·17-s − 4.16·19-s + 0.156·21-s − 1.07·23-s + 25-s − 5.61·27-s + 8.94·29-s − 0.635·31-s + 4.75·33-s + 0.103·35-s + 5.32·37-s − 2·39-s + 5.23·41-s + 8.28·43-s − 0.696·45-s + 9.86·47-s − 6.98·49-s − 3.86·51-s + 8.54·53-s + 3.13·55-s + ⋯ |
| L(s) = 1 | + 0.876·3-s + 0.447·5-s + 0.0390·7-s − 0.232·9-s + 0.944·11-s − 0.365·13-s + 0.391·15-s − 0.617·17-s − 0.955·19-s + 0.0342·21-s − 0.224·23-s + 0.200·25-s − 1.07·27-s + 1.66·29-s − 0.114·31-s + 0.827·33-s + 0.0174·35-s + 0.875·37-s − 0.320·39-s + 0.817·41-s + 1.26·43-s − 0.103·45-s + 1.43·47-s − 0.998·49-s − 0.541·51-s + 1.17·53-s + 0.422·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.921469592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.921469592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 7 | \( 1 - 0.103T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 + 4.16T + 19T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 + 0.635T + 31T^{2} \) |
| 37 | \( 1 - 5.32T + 37T^{2} \) |
| 41 | \( 1 - 5.23T + 41T^{2} \) |
| 43 | \( 1 - 8.28T + 43T^{2} \) |
| 47 | \( 1 - 9.86T + 47T^{2} \) |
| 53 | \( 1 - 8.54T + 53T^{2} \) |
| 59 | \( 1 - 9.58T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 + 0.265T + 67T^{2} \) |
| 71 | \( 1 - 2.42T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 1.45T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.367827465123727105824957099274, −7.63320232523072218050480039391, −6.70407265517974457862263802451, −6.22340250490550564792797786823, −5.32479228029210699216794340256, −4.31625105663016305580884975014, −3.78074361362253478330514976963, −2.55912454340448730220927197728, −2.26204521631430816005964662059, −0.894703658462995134499571033242,
0.894703658462995134499571033242, 2.26204521631430816005964662059, 2.55912454340448730220927197728, 3.78074361362253478330514976963, 4.31625105663016305580884975014, 5.32479228029210699216794340256, 6.22340250490550564792797786823, 6.70407265517974457862263802451, 7.63320232523072218050480039391, 8.367827465123727105824957099274
